Synchr onization and W a v es in Confined
Complex Acti v e Media
v or gele gt v on
Jan Frederik T otz
Master of Science (M. Sc.)
geb . in Berlin
v on der F akultät II – Mathematik und Naturwissenschaften
der T echnischen Uni versität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
— Dr . rer . nat. —
genehmigte Dissertation
Promotionsausschuss:
V orsitzender: Prof. Dr . Dieter Breitschwerdt
Gutachter: Prof. Dr . Harald Engel
Gutachterin: Prof. Dr . Katharina Krischer
Gutachter: Prof. Dr . Dr . h.c. Eckehard Schöll, PhD
T ag der wissenschaftlichen Aussprache: 22. Dezember 2017
Berlin 2018

Ackno wledgements
Prof. Harald Engel
Prof. Oli ver Steinbock, Prof. K enneth Show alter ,
Prof. Vladimir K. V anag, Prof. Jörn Dunkel
A G Engel (Berlin)
Stef fen Martens, Julian Rode, Dumitru Calugaru, Jakob Löber , Markus Radszuweit,
Dirk K ulawiak, Florian Buchholz, Christopher Mik olajzak, Leander Self, Fabian P aul,
Franziska Böhme, Ingebor g Gerdes, Hermann Brandtstädter , Grigory Bordyugo v ,
Jan Schlesner , Andrea Schulze, Julia Eckert, Y ulia Jagodszinski, Peter Orlo wski
A G Steinbock (Florida)
Zulma Jiménez, Sumana Dutta, Laszlo Roszol, Rabih Makki
A G Showalter (W est V irginia)
Darrell Collison, Desmond Y engi, Razan Snari, Simba Nk omo, Mark T insley
A G V anag (Kaliningrad)
P a vel Smelo v , Iv an Proskurkin, Dmitry Safonov
Special Thanks
Geor g Engelhardt, Enrico Dietz, Norbert Zielinski, Rinaldo August, Sven-Uwe Urban,
Jörn Six, F abian Sielaf f, Chris Scharfenorth, Sascha Gerlof f, Anna Zakharov a,
Madeleine Nuck, Annette T aylor , Delora Gaskins, V alentin Flunkert, Christian Hennig,
Franz-Josef Schmitt, Philipp Strasber g, Alice Schwarze, F achreddin T abataba-V akili,
F arsane T abataba-V akili, Ramona Rothfischer , Julien Olck, Sophie Seidenbecher ,
Sophie Ernst, Pirmin K ustin
Funding
Prof. Holger Stark (GRK 1558) and Prof. Eckehard Schöll (SFB 910)
Flora high school (South Car olina)
Physics teacher Mr . T om Sunday
F amily
Cordula & Claus-Dieter & Carl Hendrik & Erika & W erner
Dedicated to my lo ving wife: Sonja

Abstract
The aim of this thesis is the study and characterization of a number of self-or ganized patterns
with potential rele vance to biological systems and be yond. T o this end we utilize the well-
established oscillating Belouso v-Zhabotinsky (BZ) reaction in chemical e xperiments as well
as numerical simulations of the underlying model equations on graphics cards.
The first part of this thesis features experiments on spiral-shaped e xcitation wa v es in a three-
dimensional oscillatory medium. Their spatiotemporal e v olution is gov erned by a circular
line singularity around which the wa ves rotate. In the absence of medium boundaries, the
singularity would contract and e v entually v anish. Due to the interaction with the boundary , the
singularity may stabilize, such that it acts far be yond its theoretical life time as an autonomous
pacemaker . The influence can be tak en into account in a semi-analytical kinematic model,
which is in good agreement with e xperiments and simulations. Related patterns of electrical
acti vity play a critical role in ventricular tachycardia, a life-threatening heart arrhythmia.
A small network of discrete BZ oscillators can support periodically spreading e xcitation
wa ves. F or a small distribution of natural oscillation frequencies, the wa ves propag ate
along the permutation symmetries. It is kno wn, that comparable electric wa ves in neuronal
networks control rh ythmic muscle contraction.
In the final part of the thesis, we verify the spiral wa ve chimera state, that was predicted
by Y oshiki Kuramoto in 2002. This particular state exhibits a coherent spiral w a ve rotating
around a core that consists of incoherent oscillators. Such patterns might play a role in
nonlocally coupled cardiac and cortical tissue as well as in the photoelectrodissociation
on doped silicon wafers and arrays of superconducting quantum interference de vices and
opto-mechanical oscillators. The e xperimental setup, that we de veloped for this purpose,
furthermore allo ws for reproducible experiments under laboratory conditions on networks
with
N > 2000
oscillators. It facilitates the free choice of netw ork topology , coupling function
as well as its strength, range and time delay , which can e v en be chosen as time-dependent.

Zusammenfassung
Die Zielsetzung dieser Arbeit ist die Untersuchung selbstor ganisierter Muster , die potentielle
Rele vanz für biologische Systeme und darüber hinaus aufweisen.
Zu diesem Zweck werden chemische Experimente auf Basis der oszillierenden Belouso v-
Zhabotinsk y (BZ) Reaktion und numerische Simulation der zugrundeliegenden Modell-
gleichungen auf Grafikkarten durchgeführt.
Im ersten T eil der Arbeit werden spiralförmige Erre gungswellen in einem dreidimensionalen
oszillatorischen Medium untersucht, die periodisch um eine kreisförmige Singularität rotieren
und dabei W ellenzüge aussenden. Ohne W echselwirkung mit der Berandung des aktiv en
Mediums, würde die Singularität kontrahieren und nach endlicher Zeit v erschwinden. Unter
Einfluss der Randwechselwirkung lässt sich die Singularität stabilisieren, so dass sie weit
über ihre theoretische Lebenszeit hinaus als autonomer Schrittmacher fungiert. Der Ef fekt des
Randes lässt sich in einem semi-analytischen kinematischen Modell berücksichtigen, welches
gut mit den Er gebnissen der Experimente und Simulationen übereinstimmt. V erwandte
Muster elektrischer Akti vität spielen insbesondere auf dem Herzmuskel eine kritische Rolle
bei der v entrikulären T achykardie, einer lebensbedrohlichen Herzrh ythmusstörung.
Auf einem kleinen Netzwerk aus diskreten BZ Oszillatoren können sich periodisch Erre-
gungswellen ausbreiten. Bei enger V erteilung der natürlichen Oszillationsfrequenzen breiten
sich die W ellen entlang der Permutationssymmetrien aus. Es ist bekannt, dass ver gleichbare
elektrische W ellen in neuronalen Netzwerken rhythmische Musk elkontraktion steuern.
Im letzten T eil der Arbeit weisen wir den v on Y oshiki Kuramoto in 2002 v orher gesagten
Spiralwellen-Chimären Zustand nach. Dabei rotiert eine kohärente Spiral welle um einen
K ern aus inkohärenten Oszillatoren. Dieses Muster könnte eine Rolle in nichtlokal gekop-
peltem Herz- und Nerv enge webe spielen, in der Photoelektrodissoziation auf dotierten
Silizumscheiben, so wie auf Gittern aus supraleitenden Quanteninterferenzeinheiten und opto-
mechanischen Oszillatoren. Der zu diesem Zweck entwickelte e xperimentelle Aufbau erlaubt
es darüber hinaus, Muster auf Netzwerken mit
N > 2000
Oszillatoren unter reproduzierbaren
Laborbedingungen zu untersuchen. Dabei lassen sich nach Bedarf Netzwerktopologie, Art
der K opplungsfunktion so wie ihre Stärke, Reichweite und Zeitv erzögerung einstellen, die
darüber hinaus auch zeitabhängig sein können.

T able of contents
1 Intr oduction 1
Intr oduction 1
2 Chapter 2 5
2 . 1 T h e o r e t i c a l B a c k g r o u n d ............................ 7
2 . 2 C h e m i c a l E x p e r i m e n t s ............................ 10
2.3 Numerical simulations and a kinematical model . . . . . . . . . . . . . . . 14
2 . 4 C h a p t e r s u m m a r y ............................... 23
2 . 5 F u t u r e d i r e c t i o n s ............................... 24
3 Chapter 3 27
3 . 1 T h e o r e t i c a l B a c k g r o u n d ............................ 28
3 . 2 E x p e r i m e n t a l R e s u l t s ............................. 31
3 . 3 S h o r t s u m m a r y ................................ 43
3 . 4 F u t u r e d i r e c t i o n s ............................... 43
4 Chapter 4 45
4 . 1 T h e o r e t i c a l B a c k g r o u n d ............................ 47
4 . 2 E x p e r i m e n t a l S e t u p .............................. 53
4 . 3 N u m e r i c a l S i m u l a t i o n s ............................ 58
4 . 4 R e s u l t s ..................................... 66
4 . 5 S h o r t s u m m a r y ................................ 80
4 . 6 F u t u r e D i r e c t i o n s ............................... 81
Conclusion 85
A ppendix A Dimension reduction of oscillators and oscillatory patter ns 87
A . 1 D i s c r e t e o s c i l l a t o r s .............................. 88

x T able of contents
A.2 Continuous oscillator fields . . . . . . . . . . . . . . . . . . . . . . . . . . 95
A ppendix B Implementations 103
B . 1 E x p e r i m e n t a l s e t u p s .............................. 103
B.1.1 Setup I: Three-dimensional acti ve medium . . . . . . . . . . . . . 103
B.1.2 Setup II: T wo- and three-dimensional acti ve medium . . . . . . . . 105
B.1.3 Setup III: Discrete oscillators . . . . . . . . . . . . . . . . . . . . . 109
B.1.4 Setup IV : Discrete oscillators . . . . . . . . . . . . . . . . . . . . . 111
B.1.5 Setup V : Discrete oscillators . . . . . . . . . . . . . . . . . . . . . 113
B.1.6 LabVIEW Control Program . . . . . . . . . . . . . . . . . . . . . 116
B.2 Numerical Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . 119
B . 2 . 1 C U D A S o l v e r ............................. 121
A ppendix C Chemistry 125
C.1 Belouso v-Zhabotinsky reaction . . . . . . . . . . . . . . . . . . . . . . . . 125
C.1.1 History & Applications . . . . . . . . . . . . . . . . . . . . . . . . 125
C.1.2 Apparent violation of the second la w of thermodynamics . . . . . . 126
C . 1 . 3 R e a c t i o n m e c h a n i s m ......................... 126
C . 1 . 4 R e d o x c a t a l y s t s ............................ 129
C . 1 . 5 P a r a m e t e r d r i f t ............................ 130
C . 1 . 6 T r o u b l e s h o o t i n g ........................... 130
C . 1 . 7 C h e m i c a l r e c i p e s ........................... 132
C.2 Ov ervie w of numerical models . . . . . . . . . . . . . . . . . . . . . . . . 133
Refer ences 141

Chapter 1
Intr oduction
“... on the shoulders of giants”
– Bernard of Chartres
The second la w of thermodynamics
1
states, that during an irre versible process the total
entropy S in an isolated system al ways gro ws:
d i S
d t > 0 . (1.1)
As time passes, matter will decay from an ordered, but improbable state to a disordered
and more probable state. Y et structures of high order , namely life forms, e xist and very
successfully so: From the microscopic bacterium Pelagibacter ubique
2
that measures just
about
5 µm
b ut makes up for the lar gest cumulated species biomass worldwide to the orders
of magnitude lar ger blue whale (Balaenoptera musculus) reaching about
30 m
in size
3
. Life
pre vails despite inhospitable en vironments that are de void of oxygen
4
, belo w freezing at
− 20 ◦ C 5 or close to boiling temperatures 6 , to just gi ve a fe w examples.
This paradox was addressed from the perspecti v e of statistical physics by Erwin Schrödinger ,
one of the pioneers of quantum mechanics, in his book "What Is Life?"
7
. He not only
ar gues for the existence of a biomolecular "asymmetric crystal" as a carrier of hereditary
information, which inspired Crick and W atts to search for the DN A double helix molecule
8
,
b ut also he makes the case that li ving matter attains an ordered state and remains in it, due to
consuming "ne gati ve entropy" from its en vironment,
d i S
d t + d e S
d t > 0 . (1.2)

2 Introduction
Put dif ferently , a li ving being is an open system, which maintains a non-equilibrium state by
entropy e xchange
d e S
with its surroundings via nutrients and waste products. This notion
was formalized mathematically by Ilya Prigogine
9 , 10
as a "dissipati ve structure", that e xists
far from equilibrium and bifurcates of f the equilibrium state.
Meanwhile, the dynamics of self-or ganized structures were elucidated from a macroscopic
vie wpoint by Alan T uring and Arthur W infree who had laid out in their respecti v e seminal
papers the basis for spatial pattern formation
11
and temporal synchronization of oscillators
12
.
Self-or ganized structures are not only restricted to life. Shortly after the first working laser
13
was b uilt, Hermann Haken successfully proposed a theory for its operation based on non-
equilibrium dynamics
14
. He and W erner Ebeling went on to found and popularize the ne w
field of "syner getics" 15 , 16 in di vided Germany and throughout the scientific world.
The common thread in all these endea v ors is self-or ganization . Its defining feature is a
structure of high, life-like order that is attained due to its o wn internal, nonlinear dynamics.
No detailed, e xternal control on the structure is required.
Its continuing success was underscored with tw o Nobel prizes. One in chemistry was
recei ved by Gerhard Ertl in 2007 for de veloping the methods of surf ace chemistry in general
and for observing pattern formation on catalytic platinum surf aces during CO oxidation in
particular
17
. The other was aw arded as recently as 2017, when the Nobel Prize in Physiology
or Medicine was shared by Jef fre y Hall, Michael Rosbash and Michael Y oung for re vealing
the basic biomolecular machinery of chronobiology in virtually all li ving things, from
bacteria, fungi to plants and animals 18 , 19 .
It was recently proposed
20 , 21
, that far a way from equilibrium the second la w needs to be
adapted
d e S
d t ⏐ ⏐ ⏐ a → b + d
d t ln ( p a → b
p b → a ) + d i S
d t > 0 (1.3)
to account for transition probabilities in and out of a gi ven non-equilibrium state. This
means that the more irre versible a process is,
p a → b ≫ p b → a
, the lar ger its associated internal
entropy production will be. Furthermore this allo ws for an engineered dissipati v e adaptation,
by placing suitable dri ving forces on the system, that will enable it to reach desired non-
equilibrium states, b ut not escape from them 22 .
Future de velopments in the field of self-or ganization can be e xpected to gi ve insight into
life-like non-equlibrium phenomena as well as lead to ne w applications in smart materials
23
and soft robotics
24
founded on neuromorphic
25 , 26
and biomimetic materials
27
, ultimately
opening the door to artificial life forms 28 , 29 and ne w forms of medical therapies 30 , 31 .

3
Figur e 1.1 | Excitation wa ve patter ns in differ ent topologies.
(a) Scroll ring in a 3d continuous
domain (b) tar get wa ve on a network with symmetry clusters (c) spiral w a ve chimera on a discrete 2d
grid.
An accessible test bed for predictions in the field of self-or ganization is the prototypical
chemical oscillator , the Belousov-Zhabotinsk y reaction
32
. Its popularity is due to it gi ving
rise to concentration traces that are strikingly similiar to the biochemical acti vity of neurons
33
.
In spatially continuous systems it supports chemical patterns similar to electrical acti vity
patterns on the heart muscle that usurp its mechanical contraction before sudden cardiac
failure 34 – 36 .
In this thesis the o verarching theme is elucidating the peculiar beha vior of excitation w a ves
as stable temporally periodic patterns on dif ferent topologies. Unlike w a v es in a fluid or
conserv ativ e solitons
37
, e xcitation wa ves require an acti ve medium, that sho ws excitability
38
and is kept f ar from equilibrium due to ener gy influx like the g ain medium in a laser
15
.
The e xcitable character manifests itself in the all-or-nothing response to an e xternal per-
turbation: Small perturbations belo w a threshold ha ve a ne gligible consequence, whereas
lar ge superthreshold perturbations lead to an e xtensi ve and unique response. This definition
is inspired by biological neurons
39 , 40
, that remain quiescent in response to small current
fluctuations, b ut emit a v oltage spike once suf ficiently perturbed. Before another perturbation
can successfully trigger an e xcursion, the system must pass through its refractory stage
and return to its rest state. Such elements, connected in a chain, support the sequential
propagation of a spik e from one end to the other: an e xcitation wa ve.
Chapter 2 is focused on the formation of a metastable pacemaker , a structure that acts as a
periodic wa ve source in three dimensions. F or parameters where stable pacemakers were
pre viously thought impossible, it is shown that the y can exist due to interaction with inherent
spatial boundaries that occur in an y real system like the myocardium.
In chapter 3 , the focus shifts to excitation w a ves on netw orks. It is found that underlying
network symmetries are intertwined with the w ay an excitation w a v e propagates ov er the

4 Introduction
network. During propagation, the nodes in each symmetry cluster synchronize, such that
the y fire concurrently .
Finally in chapter 4 , a lar ge grid of discrete oscillators that are nonlocally coupled to each
other gi ves rise to a structure that was predicted 15 years ago by Y oshiki K uramoto theo-
retically
41
, b ut ne v er verified e xperimentally: the spiral wa ve chimera. The distinguishing
feature is that a coherent e xcitation wa ve rotates around a central re gion that consists of
seemingly incoherent oscillators e ven though all oscillators are coupled identically . This is
the first e xperimental example for the co-e xistence of ordered and disordered phases in a
rob ust and reproducible real setting without sensiti ve dependence on initial conditions.

Chapter 2
Confined Scr oll Rings

6 Chapter 2
Figur e 2.1 | Model f or spiral wa ve f ormation.
A forest consisting of trees that can catch fire, b urn
and regro w (unnaturally) fast is a minimal model for an e xcitable medium. (a) An e xcitation wa ve in
the form of a planar flame front tra vels from left to right. In the back of the flame front, new trees are
already gro wing. A rain cloud located in the center of the medium pre vents flames from spreading in
the top half, which breaks the front and creates an open end. (b) While the flame front propagates in
the bottom half, the rain cloud v anishes. The broken flame front continues onward to the right and
no w also spreads in the upw ard direction. This is the genesis of the spiral wa v e. (c) After a suf ficient
number of rotations, the full-fledged spiral wa ve pattern is observ able. This cellular automaton-like
simulation was realized with the Star Craft II map editor 42 .
Among the dif ferent spatio-temporal patterns of propagating excitation w a v es, rotating spiral-
shaped wa ves are v ery common. This suggests, that their emergence must depend on general
rules, that transcend microscopic details. As depicted in Figure 2.1 , an initially planar wa ve
front can break up due to interaction with inhomogeneities in an acti ve medium. The resulting
wa ve features an open end f ar from an y boundaries. Still, the e xcitation (fire) will spread from
the current e xcited region to an y surroundings that are not in their refractory , unexcitable
(b urnt) state. This means that the main front will continue forward, but at the w a ve tip
the e xcitation can spread upwards in addition. While the tip continues on its pirouette-like
motion, it becomes the source of the excitation w a ves, that are periodically emitted into
the medium. In this sense, the tip is the localized organizing center
43
of the delocalized
spiral wa ve, that has a w a v elength
λ
and rotation period
T
. Note that the oscillations at each
location outside the spiral core are entrained to the rotation period of the spiral wa ve. Here,
the non-equilibrium character manifests itself in the influx of e xternal ener gy , that is required
to return oscillators to their rest state (unb urnt) so a neighboring excitation can restart the
oscillation c ycle.
Apart from this simple e xample, spiral wa v es ha ve been observ ed in an astonishing variety of
biological, chemical and physical systems
a
: myocardium
62 – 66
, cardiac cell monolayers
67 , 68
,
optogenetically engineered cell layers
69 – 71
, giant honey bee colonies
72
, mammalian cortex
73
,
a
Some v ortex patterns in nature ha ve spiral shape, but there is no underlying e xcitable medium. Some
examples range from Bose Einstein condensates
44
, spin spirals
45
, spiral crystal growth
46 – 48
, drop-induced
airflo ws
49
, fluid mixing
50
, Bernard con vection in gas mixture
51 , 52
, Faraday e xperiments with vertically vibrated
viscous fluid and sand
53 , 54
, Kármán v ortex street
55
, hurricanes
56
, extraterrestrial storms, such as the Great Red
Spot
57
and polar v ortices
58
on Jupiter , galaxy formation
59
and the recent binary black hole as well as neutron
star mer gers, which emitted double spiral gravitation w a ves 60 , 61 .

2.1 Theoretical Background 7
calcium wa ves in frog e ggs
74
, chicken retina
75
, single cells and conglomerates of the social
amoeba Dictyostelium discoideum
76 , 77
, geographical tongue inflammation
78
, migratory
erythematous lesions
79
, human cro wds
80
, synthetic somatic cell sheets
81
, uterus during
labor
82 , 83
, yeast glycolysis
84
, Min proteins
85
and actin-proteins in electro-fused cell mem-
branes
76 , 86
, lichen gro wth
87
, cilia arrays
88
, Belouso v-Zhabotinsky (BZ) reaction medium
89
,
BZ-surfactant mixtures
90
, BZ-infused Langmuir monolayers
91
, precipitation reactions
92
, CO
oxidation on Pt-110 surface
93
, electrodeposition of a binary alloy
94
, corrosion
95
, lasers
96 – 98
,
comb ustion
99
, liquid crystals
100 , 101
, plasma
102
, dielectric barrier gas-dischar ge
103
, trapped
ultracold ion arrays
104
, optomechanical oscillator arrays
105
, coupled map lattices
106
, and
Josephson junction arrays 107 .
Among the most striking e xamples are spiral wa v es on the heart muscle, where they are
also called rotors or reentry . During the healthy operation of the heart, the sinoatrial node
entrains the beating rhythms of all cells in the myocardium by emitting a propagating electric
e xcitation wa ve. Upon arri v al at a heart cell, it expands and shortly after contracts again. This
synchronized, collecti ve mechanical deformation underlies the vital heart beat that pumps
blood through the body
108
. Due to e.g. une xcitable
109
or highly e xcitable
110
obstacles,
spiral wa ves may nucleate. Since they are periodic w a ve sources, the y directly compete with
the sinoatrial node for the entrainment of the myocardium. If multiple spiral wa ves form,
the ensuing wa ve chaos compromises the synchronized, collecti ve deformation and leads to
life-threatening fibrillation 36 , 65 .
Due to the three-dimensional nature of the heart, the rotating patterns are actually scroll
wa ves, which are made up of spiral w a ves stack ed up on each other . Accordingly , the center
of the spiral core is e xtended to a line first identified by W infree as or ganizing center of the
scroll wa ve and named filament 87 , 111 – 114 .
One of the first reported e xperimental observ ations of a filament in the cardiac muscle, had
a ring-shaped form
115
. Indeed, connecting both ends of the filament and bending it into
a ring, leads to a simple structure called the scroll ring
87 , 116
(see figure 2.2 a). Curiously ,
the periodic wa ve pattern of scroll rings w as exploited in oil-immersed BZ drops to dri ve
autonomous periodic motion via changes in the surface tension 117 .
The dynamics of this structure in the presence of boundaries are the focus of this chapter .
2.1 Theor etical Backgr ound
The equations of motion for filaments were first deri ved in a mathematically rigorous way
by James K eener in 1988 employing techniques from singular perturbation analysis
118
. His
approach is re viewed briefly in appendix A.2 together with recent de v elopments.

8 Chapter 2
Figur e 2.2 | Unperturbed dynamics of an axisymmetric, untwisted scroll ring .
(a) A scroll ring
is a toroid formed from a spiral wa ve. The tip of each spiral wa ve is attached to a ring with radius
R
,
called the filament (red). F or clarity , here only a single isosurface of the continuous concentration
pattern is sho wn. (b) W ithout perturbations the scroll ring will either decay for filament tension
α > 0
(blue) or gro w for
α < 0
(yello w), see
( 2.1 )
. T ime and space are scaled relativ e to the rotation period
T and wa velength λ of the wa ves emitted from the ring.
In the case of a closed filament loop, a scroll ring, the equations of motion simplify to
d R
d t = − α κ , (2.1)
d z
d t = β κ . (2.2)
While the ring drifts along its symmetry axis in
z
-direction, the radius
R
of the ring changes
depending on the ring curv ature
κ = 1 / R b
. It will either contract and v anish or expand
depending on the sign of the filament tension
α
, see Fig. 2.2 b . In its later stages a gro wing
ring will under go the neg ati v e line-tension instability
120
. This means it will break apart and
de velop into a highly disor ganized pattern kno wn as W infree turb ulence
121
. The analytical
solution to the radius dynamics ( 2.1 ) is gi ven by
R 2 ( t ) = R 2
0 − 2 α t . (2.3)
Thus unlike its tw o-dimensional counterpart, the spiral wa ve, a scroll ring is not a stationary
pattern. Intuiti vely the non-stationarity is clear: In two dimensions, a spiral wa ve will drift
under an e xternal perturbation. This can be realized in the form of a time-periodic e xternal
parameter field
122
, a parameter gradient
123
, an applied electric field
124
, feedback control
125
b
Note that this is a special case of an arbitrary closed curve
r
moving under mean curv ature flo w:
r t = − α κ N , where N is the normal v ector at each point of the curve r 119 .

2.1 Theoretical Background 9
or a Neumann boundary
126 , 127
. These perturbations lead to a periodic modulation of the
e xcitability in the core region. Ho wev er , in three dimensions no external field is necessary ,
since the periodic perturbation is due to the pattern itself. During one rotation period, the
filament loop will c ycle between a large and a small radius, which hav e a small and lar ge
curv ature, respectiv ely . In the same way the eikonal equation,
c n = c 0 − D κ , (2.4)
dictates ho w non-planar excitation wa ves in tw o dimensions straighten out
128 – 130
, the rotation
speed of the filament increases when it goes from a lar ge circumference to a small one and
decreases on its way back
131
. This periodic curv ature-induced self-modulation forbids
stationary scroll ring patterns.
Ho wev er , the dynamics of the unperturbed scroll ring can be complemented by considering
e xternal perturbations, specifically the ef fect of Neumann boundaries that naturally limit the
spatial e xtent in a realistic setting.
T o this end it is important to first understand how tw o-dimensional spiral wa ves interact
with Neumann boundaries. One approach is based on the con volution of perturbations with
spiral wa ve response functions
132 – 134
(appendix A.2 ). While it is highly successful for weak
perturbations, such as spatiotemporal parametric inhomogeneities, it is not applicable to the
interaction with Neumann boundaries. Boundaries break the symmetries of the Euclidean
plane strongly and thus the Goldstone modes, whose adjoints are the response functions, do
not e xist.
An alternati ve data-dri ven approach is described in the ne xt section.

10 Chapter 2
2.2 Chemical Experiments
Figur e 2.3 | Experimental setup f or boundary stabilized scroll rings.
(a) P attern formation in the
acti ve medium is monitored spectrophotometrically . Excitation wa ves of oxidized catalyst absorb less
light than the domains with reduced catalyst and thus the concentric wa ves of a scroll ring appear
in bright blue on an orange background, see photograph in (b). The white bar corresponds to
1 cm
.
(c) Absorption spectra of the reduced form of the catalyst (Fe
3+
) in orange and the oxidized form (Fe
4+
)
in blue.
While the Belouso v-Zhabotinsky reaction has been utilized to study chemical oscillations in
time, it is also readily employed to study propagating patterns in a continuous medium in time
as well as space (see appendix C.1.3 ). The applicability of this approach is well-established
by a lar ge body of experiments on scroll rings in BZ media, testing their existence
116
,
formation
135
, dynamics for positi v e and neg ati v e filament tension
136 , 137
, beha vior influenced
by each other
138
, temperature and electrical gradients
139 , 140
as well as defect sites
141 – 143
.
Ho wev er , their interaction with medium boundaries is less well explored.
Here, chemical wa ves in the acti ve medium are observ ed with a simple spectrophotometric
setup
144
(see Fig. 2.3 a). Excitation wa ves manifest themselv es as narrow , propagating
domains of oxidized catalyst surrounded by lar ge regions of reduced catalyst. Since the
absorption spectrum of the ferroin catalyst, Fe(o phen)
3
, depends on the oxidation state of
the Fe-ion (Fe
3+
/Fe
4+
), it is possible to optically record the concentration wa ves with a CCD

2.2 Chemical Experiments 11
Figur e 2.4 | Scroll ring nucleation.
Reproducible initialization of scroll rings consists of four stages;
(a) A silv er wire is introduced at the liquid-gel interface of the acti ve medium to (b) initiate a spherical
wa ve. (c) By shaking the medium strongly enough to erase an y structures in the upper liquid layer , (d)
the spherical wa ve is con verted into a scroll ring.
camera. W a v es appear as bright blue on an orange background (Fig. 2.3 b), because the re-
duced catalyst strongly absorbs blue and green parts of the VIS spectrum (
λ = 380 − 560 nm
),
whereas the oxidized form is nearly transparent to all wa velengths, e xcept for a weak ab-
sorption of red wa velengths with
λ = 580 − 620 nm
(Fig. 2.3 c). The optical contrast is
further enhanced with a bandpass filter that only transmits light into the camera with a
wa velength of
λ = 400 − 500 nm
. Further details on the experiment instrumentation are
gi ven in appendix B.1.1 .
The chemical medium consists of two layers to allo w for a reproducible nucleation protocol
of scroll rings
141
(Fig. 2.4 ). On the solid BZ reagent-infused hydrogel layer rests another
layer of liquid BZ solution. The agarose hydrogel is solid enough to pre vent con vecti ve
instabilities and also suf ficiently porous, such that the dif fusion of chemical species in both
layers is approximately identical. The first step of the initiation procedure is placing the
clean end of a pure silv er wire (
99 % Ag
) at the gel-liquid interface for about
15 s
. Adding
silv er to the BZ solution allo ws for the formation of small amounts of AgBr
145
. This locally
perturbs the chemical balance due to the remo v al of the inhibitor Br
–
. Thus shortly after the
silv er wire is remov ed, a spherical wa v e starts to expand from the former location of the wire
end. Similar to the initiation of a spiral wa v e (Fig. 2.1 ), a spherical w a ve is cut by strongly
shaking the Petri dish, which holds the chemical medium. This remov es any pattern in the
upper liquid layer . The structure in the bottom solid part remains intact. After the liquid
returns to rest, the wa ve resumes its propagation into the upper layer and forms the scroll
ring.
The beha vior of scroll rings is analyzed via space-time plots also kno wn as k ymographs.
The y are generated by plotting the intensity v alues of pix els taken along a line through the
ring center from each image of the camera image sequence collected during the e xperiment.
The space-time plots are processed further with a gradient filter to e xtract a parametrization

12 Chapter 2
Figur e 2.5 | Experiments on boundary-influenced scr oll ring dynamics.
Space-time plots ex-
tracted from a horizontal line through the center of each ring sho w e xcitation wa v es (grayscale) and
filament dynamics (red) of representati ve cases: (a) collision of the scroll ring with the bottom bound-
ary , (b) contracting free scroll ring outside the interaction range of the boundaries, (c) contracting,
boundary-influenced scroll ring with reduced contraction rate, (d) quasi-stationary scroll ring, (e)
expanding scroll ring. The vertical space axis corresponds to
2 . 3 cm
in all cases. The horizontal time
axis spans (a)
50 min
, (b)
3 h
, (c)–(e)
4 h
. The heights of the solid layer
h 1
and of the liquid layer
h 2
were (a)
3 mm
and
4 mm
, (b)
4 mm
and
4 mm
, (c)
4 mm
and
3 mm
, (d)
4 mm
and
2 . 5 mm
, (e)
4 mm
and
1 . 25 mm
. (f) - (g) T wo camera images sho w the expansion of the circular filament from (e) o v er
the course of
3 h
. The length of the white bar corresponds to
5 mm
. (h) Radius dynamics extracted
from space-time plots (a)-(e). Open circles: (a), full circles: (b), triangles: (c), full squares: (d) and
open squares: (e). Error bars result from the measurement error of the spatial filament location.
for all wa ve trains. The intersection points of consecuti ve inw ard and outward tr av eling
wa ves allo w for the calculation of the filament radius o ver time
R ( t )
. While a manual v ariant
of this procedure was used in the past
146
, it has been automatized for this thesis to analyze a
lar ge body of experiments in a short time.
The e xperiments re veal that the bottom Neumann boundary has a profound impact on the
dynamics. Before summarizing the results of the experiments, note that in all presented
e xamples, special caution has been gi ven as to detect an y occasions of filaments pinning to
CO
2
b ubbles. Experimental runs where this occurred were discarded, as pinning is kno wn to
lead to an increase in lifetime as well as stabilize filaments
141 , 147
. Successful experiments

2.2 Chemical Experiments 13
without defect interactions can be subdi vided into fi ve qualitati vely dif ferent categories based
on their dynamics.
In the e xtreme case of initiating a scroll ring close to a boundary , the ring v anishes after fe w
periods (Fig. 2.5 a). Instead of contracting, the ring collides with the boundary . The collision
does not happen concurrently along the filament: First a small fraction of the line singularity
disappears and one rotation later the rest follo ws. Note that the filament gro ws slightly before
colliding (Fig. 2.5 h).
Suf ficiently far aw ay from an y boundaries,
d = 0 . 7 λ
, rings with positi ve filament tension
α
simply contract follo wing equation
( 2.1 )
, see figure 2.5 b . In this case the squared radius
decreases approximately linear with time in agreement with equation
( 2.3 )
, as can be seen
in Figure 2.5 h. The measurement of the filament tension yields
α = 0 . 026 λ 2 T − 1
, which is
in good agreement with pre vious experiments
148
. As can be expected its measured lifespan
(2 . 6 hours) is in good agreement with its predicted lifetime
t L = R 2
0
2 α ≈ 2 . 5 hours . (2.5)
At a distance of
d = 0 . 5 λ
, the boundary influence already impedes the natural contraction
(Fig. 2.5 c). This leads to a lar ger lifespan (
2 . 8
hours) in comparison to what would ha ve
been e xpected (
2 . 0
hours) based on the initial radius
R 0
of the ring. Even though its initial
radius is smaller , it li ves significantly longer than the unperturbed ring (see yello w and blue
markers in Fig. 2.5 h).
In a small interv al of boundary distances
d
, the contraction rate of rings ef fecti vely v anishes
(Fig. 2.5 d). The radius
R
barely changed for
55
periods, lasting ov er
5
hours. Longer
measurements were inconclusi ve due to widespread nucleation of CO
2
b ubbles throughout
the acti ve medium. Ho wev er , gi ven that the ring w as e xpected to disappear after
2 . 9
hours due
to contraction in the absence of boundaries, its persistence distinguishes it as a self-or ganized
pacemaker .
Une xpectedly , scroll rings with positi ve filament tension,
α > 0
, may also e xpand under the
influence of a Neumann boundary (Fig. 2.5 e-g). This behavior w as pre viously assigned to
ne gati ve filament tension,
α < 0
, e xclusi v ely
149
. Expected to v anish after
1 . 8
hours, the ring
radius is at
7 . 2 mm
after
3 . 0
hours and continues to e xpand with
0 . 5 mm h − 1
. During this
time the ring radius e xpanded by
40 %
. Also, the ring does not sho w angular deformations
during the e xpansion (Fig. 2.5 f-g). Regular and deformed black circles correspond to CO
2
b ubbles that nucleated in the liquid and gel part, respectiv ely . Ho we ver , no pinning interaction
between b ubbles and filament was observ ed.

14 Chapter 2
The last two cases strikingly illustrate that a Neumann boundary does not lead to a small
alteration of the contraction, b ut dramatically changes the e xpected behavior .
2.3 Numerical simulations and a kinematical model
Figur e 2.6 | P erturbations to confined small scroll rings.
(a) One half of a cross section through a
scroll ring is a spiral wa ve. (b) For a small scroll ring radius
R
, the spiral wa ve on the opposite site of
the ring induces a perturbation that is equi v alent to one from a Neumann boundary at the symmetry
axis of the ring. Like wise when the scroll ring is close to a real bottom Neumann boundary , the
constituting spiral wa ves are perturbed by it. In both situations the boundary causes spiral wa ve drift.
Ho w exactly the observ ed surprising dynamics (Fig. 2.5 ) emerge is in vestigated further in
numerical simulations. Especially the apparent contradiction of an expanding ring despite
positi ve filament tension is resolv ed by constructing a detailed, kinematical model that
accounts for Neumann boundaries.
From numerical simulations of three-dimensional e xcitable media, it is expected that bound-
aries may stabilize v orte x patterns in systems with neg ati v e line tension
α < 0 150 , 151
. This
was also recently confirmed e xperimentally with setup II
152
(see appendix B.1.2 ). Intuiti vely
this beha vior is clear . An inspection of the spatial setting of the bounded scroll ring re veals,
that it is equi v alent to a two-dimensional spiral w a ve drifting at a Neumann boundary , since
one half of a cross section through the ring is a spiral w av e (see Fig. 2.6 ). Furthermore, it is
well-established theoretically and e xperimentally that Neumann boundaries in two dimen-
sions cause a stable spiral wa ve drift at a fix ed distance in parallel to the boundary with a
constant v elocity
c ∥
due to resonant perturbations
126 , 132 , 153
. A simple dynamical system
for the beha vior of a scroll ring in boundary proximity takes this drift attractor into account.

2.3 Numerical simulations and a kinematical model 15
Figur e 2.7 | Numerically determined drift-velocity field components.
(a) T rajectories of spiral
wa ves in the Neumann boundary-induced drift v elocity field. The trajectories of the core centers (red)
are extracted from a veraging o ver the spiral tip trajectories (blue). The size of the gray arro ws is
scaled monotonically to visualize the direction and modulus of the v ector field across dif ferent orders
of magnitude. (b) Both velocity components are plotted o ver distance. The normal component has
two fix ed points. The v elocity at the stable fixed point f ar from the boundary ,
z a t t ≈ 0 . 824605 λ
, is too
small to cancel ring contraction. The unstable fixed point at
z re p ≈ 0 . 2834785 λ
, is close enough to
the boundary to ex ert a potential influence on ring dynamics. The solid lines through the measurement
points are high-order spline interpolating functions.
Augmenting equation ( 2.1 ) leads to
d R
d t = − α / R + c ∥ . (2.6)
The fix ed point exists if the intrinsic radius dynamics and drift motion cancel each other . For
the case of ne gati ve line tension
α < 0
this fix ed point is stable. Howe ver , linear stability
analysis for positi ve line tension re veals an unstable fix ed point, so the stationary dynamics
observ ed in the experiment (Fig. 2.3 d) was not thought to be possible for α > 0.
T o resolve this conundrum, a more detailed measurement of the boundary-induced drift
v elocity field
c ( z )
outside of the stable drift attractor is required. Numerical simulations
can be employed to easily e xclude an y unwarranted perturbations on the spiral wa ve b ut the
boundary . The chemical kinetics of the ferroin-catalyzed BZ reaction are well-described by
the Ro vinsky model 154 (see appendix C.2 ):
∂ u
∂ τ = 1
ε [ u ( 1 − u ) − ( 2 qa v
1 − v + b ) u − µ
u + µ ] + D u ∇ 2 u , (2.7)
∂ v
∂ τ = u − a v
1 − v + D v ∇ 2 v . (2.8)

16 Chapter 2
T able 2.1 | V ortex pr operties.
Comparison of characteristic properties of e xperimentally and numer -
ically observed v ortex patterns.
property e xperiment simulation
wa velength λ 5 . 8 mm 5 . 6 mm
rotation period T 390 s 372 s
filament tension α 0 . 026 λ 2 T − 1 0 . 024 λ 2 T − 1
The parameters of this reaction dif fusion system are deriv ed from concentrations used in
the chemical e xperiments (see table C.4 ). Resulting pattern characteristics from numerical
simulations, such as wa velength, rotation period and filament tension, are in excellent
agreement with the e xperiments. A comparison of the characteristic properties is gi ven in
table 2.1 .
The drift v elocity field induced by a planar Neumann boundary
c = ( c ∥ , c ⊥ )
is measured
in simulations of a spiral wa ve in a tw o-dimensional domain, see Fig. 2.7 . The spiral
tip trajectory is e valuated in such a w ay , that the tip rotation is av eraged out and only the
trajectory of the core center remains. After a transient these trajectories are analyzed to
reconstruct the drift v elocity field in dependence of the distance to the boundary . As can
be seen in figure 2.7 a, the stable drift attractor is so far a way from the Neumann boundary ,
z a t t ≈ 0 . 82 λ
, that the spiral wa ve motion is barely af fected. All sho wn trajectories are
inte grated ov er the same time duration of
300
unperturbed rotation periods. There is also
another attractor at
z re p ≈ 0 . 28 λ
, b ut it is unstable. An y spirals that start very close to it, will
either be repelled a way from the boundary or pushed into it. Due to the translation symmetry
of the drift field,
c ( x , z ) = c ( z )
, its components can be plotted ov er the distance
z
from the
boundary (Fig. 2.7 b).
No w that the complete boundary-induced drift field is known, we can construct a detailed
kinematic model. T o this end we will take into account the real bottom Neumann boundary
as well as the symmetry axis. Unless the ring is v ery lar ge, each spiral wa ve feels its opposite
counterpart. Since the configuration is mirror symmetric about the z-axis, the concentration
gradient across the axis v anishes:
∇ u = ∇ v = 0
. The same condition holds true at a Neumann
boundary (see Fig. 2.6 b). Incorporating both boundaries into the kinematical model of a
free ring,
( 2.1 )
and
( 2.2 )
, results in a semi-analytic model that accounts for the boundary

2.3 Numerical simulations and a kinematical model 17
Figur e 2.8 | V isualization of a simulated scroll ring .
All parts of the concentration field
u ( x , y , z )
are colorized in blue whose v alues are lar ger than 0.35. The red ring highlights the position of the
filament, which is the organizing center of the entire ring. A fraction of the ring is transparent to
visualize the inner parts.
influences:
d R
d t = − α / R + c ∥ ( z ) + c ⊥ ( R ) , (2.9)
d z
d t = β / R + c ⊥ ( z ) − c ∥ ( R ) . (2.10)
V ariables
R
and
z
still describe the radius of the ring and its position along the symmetry axis.
The influences of a horizontal Neumann boundary at
z = 0
and a virtual v ertical Neumann
boundary at
R = 0
(see Fig. 2.6 ) are introduced as additi ve perturbations. The perturbations
are the pre viously measured boundary-induced drift velocity fields in tw o dimensions
c ( z )
.
The minus sign in front of
c ∥ ( R )
accounts for the z-axis being oriented in the opposite
direction in comparison to the setting it was measured in (Fig. 2.7 a). The distance to the
boundary is not af fected by the flip, so c ⊥ ( R ) is included with a positiv e sign.
The v alidity of the modified kinematical model is checked by comparing solutions of the
two-component ODE
( 2.9 )
with the filament dynamics e xtracted from the time e v olution of a
full three dimensional partial dif ferential equations,
( 2.7 )
and
( 2.8 )
, that faithfully reproduces
the chemical dynamics in the Petri dish (see section 2.2 ). c
c
Note that the azimuthal symmetry of the scroll ring could be e xploited for fast simulations in a two-
dimensional domain of reduced cylindrical coordinates
155
. Ho wev er , this approach would also suppress
potential instabilities along the azimuthal direction. For e xample, it is known, that an inclined orientation of the
filament plane relati ve to the Neumann boundary induces twist w a ves along the filament 156 , 157 .

18 Chapter 2
The reaction dif fusion equations are solved on a three-dimensional Cartesian grid employing
a finite-dif ference explicit Euler scheme
158
for speed reasons with a time step of
∆ t = 0 . 05 s
.
The dif fusion operator is ev aluated ov er a 7-point Laplacian stencil with
∆ x = 0 . 001 cm
. The
instantaneous filament location is determined as the set of points, which form the intersection
of two le vel sets of the acti v ator v ariable
u ( x , y , z ) = 0 . 2
from subsequent time steps. The
simulation domain is a lar ge cuboid with length and width of
6 λ
and height of
1 . 5 λ
. The
b ulk v olume is bounded by Neumann boundaries. The spiral wa v e length
λ
is measured in
adv ance to compute the required lengths. The scroll rings in the simulations are initiated in
a similar manner to the e xperiments: They nucleate from an initial c ylindrical wa ve whose
upper fraction is remo ved at the start of the simulation. The size of the deleted upper fraction
determines the initial distance
z ( t = 0 )
to the interacting, bottom Neumann boundary . The
top and lateral boundaries are placed suf ficiently far aw ay to ne glect their influence on the
ring. A fully dev eloped scroll ring is sho wn in figure 2.8 . The three-dimensional numerical
simulations are parallelized for fast computation and are performed on a graphical processing
unit (Nvidia GTX 970), more implementation details and a speed comparison are in the
appendix B.2 .
As can be seen in figure 2.9 a, the observ ed filament dynamics of the PDE,
( 2.7 )
and
( 2.8 )
, in
three spatial dimensions are in e xcellent agreement with the solutions of the semi-analytical
kinematical ODE model
( 2.9 )
. Thus the influence of the boundaries is correctly incorporated
in the reduced ODE model
( 2.9 )
. T o facilitate the comparison with the e xperimental results,
distincti ve numerical trajectories are labeled (A-E) corresponding to the subfigure labels (a-e)
in figure 2.5 . All cases observ ed in the chemical experiments were reproduced in numerical
simulations of the underlying reaction dif fusion equations and accounted for in the modified
kinematical model ( 2.9 ).
As in two dimensions (Fig. 2.7 a), rings starting at a distance of
z > 0 . 5 λ
are barely af fected
by the bottom Neumann boundary (case B) and contract according to the unperturbed
dynamics
( 2.1 )
. Only to wards the end of their lifetime
t L
, when they reach a radius of
R < 0 . 5 λ
the y are slightly pushed do wnwards, which is due to the mirror symmetry of the
ring.
The repelling line in the drift v elocity field at
z ≈ 0 . 28 λ
still plays an important role. It
separates rings that v anish by collision with the Neumann boundary at
z = 0 λ
(A) from those
that contract
R → 0 λ
(C). The contraction rate of repelled rings is impeded initially by the
antagonistic influence of the boundary-induced drift
c ∥ ( z )
. Ho we v er , due to the repulsion,
the impeding influence of the boundary lessens with increased distance from it (Fig. 2.7 b).
W ithin the framew ork of the kinematic model
( 2.9 )
it is also possible to e xplain the unex-
pected observ ation of persistent and expanding scroll rings in the e xperiment (Fig. 2.5 d,e).

2.3 Numerical simulations and a kinematical model 19
Figur e 2.9 | Agreement between full thr ee-dimensional simulations and kinematical model.
(a) Radius
R
and boundary distance
z
of the three-dimensional filament e v olution (blue) are ex-
tracted to compare with the solutions (red) of the kinematical model
( 2.9 )
. T rajectories highlighted
with capital letters (A-E) correspond to the fi v e distinct cases observed in the e xperiment, which are
summarized in figure 2.5 (a-e). At the intersection of the
R
- and
z
-nullclines (dashed and solid) is an
unstable fixed point, which is mark ed by an unfilled circle. (b) A space time plot of the quasi-stationary
solution (D) sho ws the long-time stable behavior of the transient. Parameters for the reaction dif fusion
system are as before and in the kinematic model α = 0 . 0245 λ 2 T − 1 and β = − 0 . 0004 λ .
Expanding scroll rings (E) are a special case of rings that annihilate at the boundary (A).
Ho wev er , they start closer to the repelling line and thus gro w for a longer time before ulti-
mately colliding with the boundary as well. Note that the dashed
R
-nullcline in Figure 2.9 a
re veals that e xpanding rings require a minimum initial radius R 0 .
Surprisingly , the fixed point in the modified kinematic model
( 2.9 )
is unstable, e ven though
e xperimental observ ation (Fig. 2.5 d) suggests otherwise. This contradiction is resolv ed
by analyzing the beha viors of scroll rings that originate in a small re gion enclosed by the
R
-nullcline from abo ve and the
z
-nullcline from belo w at large radii (D). Before entering
the contraction phase, these rings under go a long-lasting transient during which their radius
barely changes. An e x emplary e v olution of such a ring during its first 40 periods is depicted
in the space-time plot sho wn in figure 2.9 b .

20 Chapter 2
Figur e 2.10 | Scroll ring lifetimes.
For dif ferent initial v alues of radius
R 0
and boundary distance
z 0
the lifetime of the contracting scroll ring is determined via the modified kinematical model
( 2.9 )
. The
peak in the lifetime plane corresponds to the quasi-stationary solution. The saturation of the lifetime
for lar ge
z 0
corresponds to the lifetime of a free ring. The flat plateau for small
z 0
corresponds to rings
that collide with the boundary .
Ha ving v alidated the modified kinematical model
( 2.9 )
, it is no w possible to run a lar ge
number of simulations in a v ery short amount of time to determine the lifetime
t L
of scroll
rings starting at dif ferent initial conditions
( R 0 , z 0 )
. For a fix ed initial distance from the
boundary
z 0
, larger rings outlast smaller rings. Rings that start at a suf ficient distance of
z 0 > 0 . 5 λ
are not af fected by the boundary . Their lifetime sho ws the e xpected quadratic
dependence
( 2.5 )
on the initial ring radius
R 0
. Initial distances
z 0
closer to the repelling line
at
z re p
lead to rings with diminished ring contraction rates that li v e slightly longer . Expanding
and quasi-stationary rings contrib ute to the peak in figure 2.10 . There, lifetimes are more
than tripled. If rings start too close to the boundary , their lifetimes are an order of magnitude
smaller in comparison to their unperturbed counterparts. This is in agreement with pre vious
e xplicit simulations 155 of the reaction dif fusion system.
The use of the modified kinematical model
( 2.9 )
is not limited to an accelerated exploration
of parameter space. It can also be used for a bifurcation analysis depending on the filament
tension
α
and drift coef ficient
β
. Note that the right hand-sides of the dynamical system
( 2.9 )
,
f R ( R , z )
and
f z ( R , z )
are not kno wn explicitly . Instead they are defined by spline interpolation
applied to the numerical measurements of the spiral w av e drift velocity field
c ( z )
(Fig. 2.7 b).
The fix ed points are found numerically as the roots of
( 2.9 )
using Ne wton’ s method
158
. T o
accelerate their detection, the initial estimate for each parameter set is based on the roots

2.3 Numerical simulations and a kinematical model 21
Figur e 2.11 | Bifurcation analysis of the semi-analytical model.
The fixed points of
( 2.9 )
and
their linear stability are e v aluated numerically , since analytic expressions are not a vailable. The
approximated right hand sides
f R ( R , z )
and
f z ( R , z )
of
( 2.9 )
are constructed from the interpolated
functions sho wn in figure 2.7 b . The bifurcation di viding unstable from stable rings is re v ealed to be a
Hopf bifurcation, due to the non-v anishing imaginary part (b), while the real part (a) of the lar gest
eigen v alue changes its sign.
found pre viously . In figure 2.11 only fixed points with
R 0 ∈ [ 0 . 1 , 2 . 0 ] λ
and
z 0 ∈ [ 0 . 1 , 1 . 0 ] λ
are taken into account. V alues smaller than
0 . 1 λ
are unrealistic, since the spiral core, the
area around which the spiral tip rotates, has a radius of about
0 . 1 λ
. At larger distances than
z 0 = 1 . 0 λ
, the boundary interaction v anishes. Also it is not feasible to initiate scroll rings
with radii lar ger than 2 . 0 λ using the setup described in section 2.2 .
In the ne xt step, the linear stability of the detected fixed points is analyzed via their eigen v al-
ues. The required Jacobian D f is e v aluated numerically using central dif ferences:
D f = ( f R ( R 0 + ∆ R , z 0 ) − f R ( R 0 − ∆ R , z 0 )
2 ∆ R
f R ( R 0 , z 0 + ∆ z ) − f R ( R 0 , z 0 − ∆ z )
2 ∆ z
f z ( R 0 + ∆ R , z 0 ) − f z ( R 0 − ∆ R , z 0 )
2 ∆ R
f z ( R 0 , z 0 + ∆ z ) − f z ( R 0 , z 0 − ∆ z )
2 ∆ z ) (2.11)
The analysis sho ws that the majority of parameter combinations
( α , β )
yield unstable fix ed
points (Fig. 2.11 ). Ho we ver , there is a small region, which features stable fixed points
for v ery small v alues of positi v e filament tension
α
and drift coef ficient
β
. Pre viously this
beha vior was purely associated with ne gati ve line tension. Both domains are connected via a
Hopf bifurcation, since the imaginary part of the lar gest eigen v alue does not v anish during
the transition.
The attracting fix ed point in the
R
-
z
phase space is due to a cusp in the
R
-nullcline (Fig. 2.12 a).
In the back of the cusp the
R
-nullcline has a ne gati v e slope and the
z
-nullcline has a positi ve
slope allo wing for a stable fixed point. The time series (Fig. 2.12 b) sho ws that the transient

22 Chapter 2
Figur e 2.12 | Time e volution of stable scr oll ring.
(a) Filament dynamics of a stable ring in the
R
-
z
phase space. Sho wn in red is the solution of
( 2.9 )
for
α = 0 . 002 λ
and
β = 0 λ
. The trajectory starts
at
( R 0 , z 0 )=( 1 . 2 , 0 . 35 ) λ
(red dot) and ends in the stable fixed point (black dot) located behind a cusp
in the dashed
R
-nullcline. (b) The radius
R
(blue) and boundary distance
z
(red) as a function of time
t
sho w the long transient period before reaching the stable fix ed point at ( R , z ) ≈ ( 0 . 49 , 0 . 48 ) λ .
is v ery long: It takes 800 rotation periods before reaching the fix ed point. For the purpose of
an e xperimental confirmation it would be adv antageous to start closer to the fixed point.
This transition also plays a role in the dynamics of scroll rings, simulated with the Ore gonator
model
( C.2 )
. Close to the bifurcation, breathing filaments exist, whose radius is a periodic
function of time 152 .
The interaction across the symmetry axis alone can also lead to stabilization of the scroll ring
(Fig. 2.13 ). After a transient of about 50 periods, the self-stabilized ring exhibits a constant
radius at
R = 0 . 38 λ
and drifts with constant speed of
v z = 0 . 13 λ T − 1
along its symmetry axis.
While there are pre vious studies of this object
134 , 159 – 161
, so far none confirmed its e xistence
in full three-dimensional simulations o ver a long period of time. Here, the simulation was
performed with the three-component Ore gonator model
( C.2 )
in a cuboid with periodic
boundary conditions on all sides. Lateral boundaries are placed at suf ficient distance to
e xclude their influence. Ne w initial conditions are required, as the cut cylinder used before
would gi ve rise to a scroll ring pair due to the periodic boundary conditions. F or this purpose,
we start from a planar e xcitation wa v e tra veling in the
z
-direction. Then a circular hole of the
desired initial radius
R 0
is cut out of the planar wa ve. Along the open edge wa ves start to
curl in and gi ve rise to the scroll ring.

2.4 Chapter summary 23
Figur e 2.13 | Self-stabilized scroll ring .
(a) The ring periodically tra verses t he simulation box with
constant velocity due to the periodic boundaries. (b) After a transient, the ring reaches its stable radius
of
R = 0 . 38 λ
. The blue and red curves are the result of a veraging o ver the fast filament oscillations
during each rotation period. (c) A snapshot of the planar filament (red) in the three-dimensional
simulation v olume. The current position of the filament is highlighted by black projections on the
box faces. The model parameters of the simulation are:
ε 1 = 1 / 8 , ε 2 = 1 / 720 , q = 0 . 002 , f = 1 . 16
,
φ = 0 . 0117
with
∆ x = ∆ y = ∆ z = 0 . 3
and
∆ t = 0 . 001
. The filament was detected as the intersection
of the isoconcentration le v el sets u ( x , y , z ) = 0 . 3 and v ( x , y , z ) = 0 . 2.
2.4 Chapter summary
Neumann boundaries transform transient scroll rings into autonomous pacemakers in a
homogeneous medium. As such they act as a self-or ganized source of periodic excitation
wa ves. Une xpectedly , this is possible e ven for rings with positi ve filament tension, that would
contract and v anish far from any boundaries. Be yond stabilizing - if only for a long transient -
Neumann boundaries can in vert the radial contraction, such that rings e xpand. This case w as
pre viously associated with negati ve filament tension e xclusiv ely .
All these distinct scenarios of scroll ring e v olution observ ed in the experiment (Fig. 2.5 )
could be successfully reproduced with numerical simulations of the underlying reaction
dif fusion equations (Fig. 2.9 a) that are in excellent agreement with a kinematical model
( 2.9 )
which is augmented to correctly account for boundary ef fects 162 .
W ith the semi-analytical, data-driv en model
( 2.9 )
v ery fast e xplorations of parameter space
are possible (Fig. 2.10 ). In addition bifurcation analysis can be performed, e ven though
the e xplicit formulas for boundary influenced scroll ring e v olution are unkno wn (Fig. 2.11 ).
This re veals the possibility of stable boundary-stabilized rings with positi ve filament tension
(Fig. 2.12 ). Last b ut not least, the e xistence of self-stabilized rings in the absence of
boundaries is also confirmed (Fig. 2.13 ).
Gi ven that three-dimensional unperturbed scroll rings disappear on their o wn, the number of
dif ferent scenarios under which persistent rings may appear is remarkable.

24 Chapter 2
2.5 Futur e dir ections
Alternati ve strate gies for the stabilization of scroll rings were pre viously in vestigated numer -
ically , but ne ver v erified experimentally . This includes parametric spatiotemporal fluctua-
tions
163
or periodic forcing
121 , 164 , 165
. Feedback methods based on the readout of localized
detectors
166 , 167
ha v e not been used so far to stabilize rings. Due to the inherent dif ficulties of
applying a spatiotemporal parametric field in three dimensions, a more ele gant method might
be to select the shape of boundaries in such a way that the y stabilize rings at a predetermined
radius.
Measuring the boundary-induced drift v elocity field in an experiment is dif ficult. The
first problem is a reliable detection of the spiral wa ve tip. Instead of detecting it by the
intersection of grayv alue-isolines from consecutiv e images, another more accurate option
is to train image recognition software on the shape of the spiral tip. The other problem is
measuring the v elocity reliably . Even in simulations it is dif ficult to e xtract v alid data points
close to the boundary , since the spiral wa ve will collide with it. T o circumv ent this problem,
one could apply an electric field. Spiral w av e are kno wn to drift in a v oltage gradient
124
with
constant v elocity
c E = ( c E
x , c E
z )
. T ogether with the boundary induced drift field, the equation
of motion for the spiral core are:
d x
d t = c ∥ ( z ) + c E
x (2.12)
d z
d t = c ⊥ ( z ) + c E
z (2.13)
After measuring the constant drift due to the electric field
c E
, it can be aligned to counteract
the component of the boundary-induced drift v elocity field
c ⊥ ( z )
that pushes spiral wa ves
a way or into the boundary , such that:
d z
d t = c ⊥ ( z ) + c E
z = 0 (2.14)
While the spiral wa ve drifts slo wly in parallel to the attractor , its drift speed
c ∥ ( z ) + c E
x
can be
measured reliably . This measurement can be repeated for dif ferent amplitudes or orientations
of the electric field, in order to counteract dif ferent strengths of the perpendicular compo-
nent
c ⊥ ( z )
. After subtracting the contrib ution from the electric field, one can reconstruct the
boundary-induced drift v elocity field for both components c ∥ ( z ) and c ⊥ ( z ) with precision.
T o further improv e upon the modified kinematical model
( 2.9 )
, it is necessary to take into
account drift v elocity functions
c ( z ; p )
that depend on system parameters
p
, as does the
filament tension α ( p ) and the drift coef ficient β ( p ) .

2.5 Future directions 25
The strate gy of stancing shapes out of a planar wa ve to nucleate a filament, can be de veloped
further . The approach is not limited to circular filaments. It is also possible to create initially
elliptic, square or heart-shaped filaments. The e volution of the latter case is interesting since
its curv ature varies from
κ → − ∞
to
κ → + ∞
while tra versing from one cusp to the othe r .
Another interesting case are yin-yang patterns, that keep their shape while rotating under
curv ature flow
119 , 168
. The simulations can be v alidated in auto-completion experiments
with a photosensiti ve BZ reaction
169
, where the top layer of an acti ve medium is uniformly
irradiated with light e xcept at the edges of the desired shape.

Chapter 3
T ar get W a v e Synchr onization on a
Netw ork

28 Chapter 3
Human brain acti vity is an enigma. There is no chance in the near future to unra vel the
network of billions of neurons and trillions of connections between them
170
. T o make matters
worse, this is only a static snapshot of a single instant in time. Discov ered more than a century
ago by one of the pioneers of neurobiology , Santiago Ramón y Cajal, the brain connectome is
not static, b ut dynamic
171 , 172
. This plasticity allo ws for learning
173 – 175
, memorization
176 , 177
and encompasses re generation 178 , 179 .
One of the currently unsolv ed puzzles of inner brain mechanisms reg ards synchronization
180
.
Experiments sho wed its relev ance in perception processing and motor control
181 , 182
, while
pathological enhanced synchronization underlies symptoms in P arkinson’ s disease
183
, T inni-
tus
184 , 185
and epilepsy
186 , 187
. In a v ery simplified description, each neuron can be reg arded
as an oscillator . Since the pioneering work of Norbert W iener
188
electroencephalographic
measurements re vealed the occurrence of synchronization among nerv e cells in the brain.
Further observ ations rev eal that neurons tend to synchronize without a time-lag or in-phase
∆ φ = 0 189
, e ven though the y are far apart and communication speed is finite
190
. One ap-
proach is, that in-phase synchronization can be relayed via an intermediate
191 , 192
, other
options are lag synchronization 193 and remote synchronization 194 .
This chapter features e xperimental results on lag synchronization with neuromorphic chem-
ical micro-oscillators. The notion of a target w a ve in a continuous medium is e xtended to
networks via an analysis of the underlying symmetries.
3.1 Theor etical Backgr ound
The last decade sa w an increased theoretical interest in generalizing continuous pattern
formation terminology to discrete networks. The groups of Mikhailo v and Schöll for example
generalized T uring
195 , 196
and Benjamin-Feir
197
instabilities as well as the eikonal equation
and propagation f ailure of excitation w a v es
198 , 199
to networks. Like wise lag-synchronization
on a network is deeply connected to propagating e xcitation wa v es in an acti ve medium. Once
the e xcitation threshold is crossed due to coupling to the surrounding medium or neighboring
oscillators, an oscillator emits a spike
38
. The spatial profile of these wa ves is determined by
initial conditions. In two dimensions the basic e xamples encompass planar , spiral and circular
wa ves, the last of which are kno wn as tar get patterns
38
. In the same sequence, the w av e
propagation of these patterns accumulates translation, rotation and reflection symmetries.
The tar get wa ve is special, because it conforms to all symmetries in the system, which
together form the Euclidean symmetry group E(2). Also note that in a suitable co-moving
reference frame, the concentration wa ves are stationary in time, c ( r , t ) = c ( r − v t ) .

3.1 Theoretical Background 29
Figur e 3.1 | Symmetries in continuous media and discrete netw orks.
(a) A tar get pattern in a
continuous BZ medium propagates, while preserving rotation (red), reflection (yello w) and translation
(blue) symmetry . (b) The symmetries on a network are permutation automorphisms highlighted by
identical edge colors. Under application of any of the corresponding permutation operators the pattern
stays in v ariant throughout its e volution. On a network, a tar get wa ve is harder to nai vely detect, since
the position of the drawn nodes does not necessarily reflect their distance form a pacemak er site. In
both subfigures symmetry operations are depicted as arro ws.
On a network of discrete nodes
200
the generalization of the tar get wa ve is not straightforw ard,
because the Euclidean symmetry group is not applicable. Instead, a network of identical
nodes features discrete permutation symmetries
201
. Permutation operations, defined by the
permutation matrix
P
, e xchange up to
N
nodes of a network. The connections between the
nodes are gi ven by the adjacenc y matrix A , whose elements follo w:
A i j = ⎧
⎨
⎩
1 , node i is connected to node j
0 , node i is not connected to node j . (3.1)
A gi ven permutation operation is a symmetry or automorphism of the network, if it obe ys
194
:
A P − P A = [ P , A ] = 0 . (3.2)
This means, that the adjacenc y matrix
A
is in variant under the gi ven permutation operation.
Furthermore, e v ery network can be completely partitioned into symmetry clusters, which
are defined as sets of nodes, whose exchange lea ves the netw ork connecti vity unaltered. A
symmetry cluster that contains only a single node is called a tri vial symmetry cluster . The
only applicable permutation operation in this case is the identity operation: It can only be
e xchanged with itself without changing the connecti vity .

30 Chapter 3
Recently Lou Pecora and co workers
202
were able to generalize their Master Stability Func-
tion (MSF) formalism
203 , 204
e xploiting network symmetries. Their formalism applies to a
network, whose N nodes are oscillators, that communicate along the edges of the netw ork:
d c i
d t = f ( c i ) + K
N
∑
j = 1
A i j H ( c i , c j ) (3.3)
The
n c
dynamical v ariables of each oscillator
i
e volv e according to the local dynamics
f : R n c → R n c
. The coupling to the network consists of two parts: The adjacency matrix
A ∈ R N × N
encodes what the neighbors of node
i
are and the coupling function
H : R n c → R n c
determines ho w node
i
is coupled to its neighbors, for example dif fusi v ely or pulse-coupled
205
.
The accumulated signals of the neighbors are scaled with the coupling strength K .
In this setting, the MSF allo ws for calculating the linear stability of the fully in-phase syn-
chronized state, where
c i ( t ) = c sync ( t )
for all
N
oscillators in the network. This is achie ved
by calculating the L yapuno v exponents of transv ersal perturbations to the synchronization
manifold. The longitudinal perturbation as the Goldstone mode has neutral stability
206
.
Ho wev er , this analysis was only applicable to the complete networ k, but not subnetw orks.
Furthermore the connecti vity of all nodes is required to be identical, otherwise the synchro-
nization manifold is not stable: Once all nodes reach the synchronized state
c sync ( t )
, the y
need to recei ve the same input from their neighbors to remain synchronized.
Incorporating symmetries allo ws for decoupling the nodes, such that the synchronization
manifolds of the decoupled symmetry clusters can be in vestigated independently . Intuiti vely ,
this approach works, because nodes in the same symmetry cluster can be exchanged without
alteration of the network. This implies that the nodes in a symmetry cluster are all coupled
identically to the rest of the network and vice v ersa. As an example, the e v olution equations
for the two orange nodes in figure 3.1 b, here in k eeping with later notation labeled as
i = 2 , 3
,
are
d c 2
d t = f ( c 2 ) + K ( 2 c magenta + c green + c cyan − 4 c 2 ) , (3.4)
d c 3
d t = f ( c 3 ) + K ( 2 c magenta + c green + c cyan − 4 c 3 ) . (3.5)
In the abo ve e xample, the coupling function was tak en as the concentration dif ference
between two coupled oscillators,
H ( c i , c j ) = c j − c i
. The coupling term for both oscillators
consists of identical contrib utions from the other symmetry clusters. Gi v en that the other
nodes are synchronized in their respecti ve symmetry clusters, all orange nodes recei ve
the same input from the rest of the network and thus can be treated as an independent

3.2 Experimental Results 31
subnetwork. While pre viously , the applicability of the MSF was limited to the stability of
global synchronization of identically coupled nodes, the symmetry decomposition allo ws for
e valuating the stability of symmetric subnetw orks, such as those found in po wer grids 202 .
Ev en introducing small parameter mismatches in the oscillators still allo ws for the successful
application of the MSF
207
. Ho we ver , at suffi ciently large parameter mismatches in-phase
synchronization ceases to e xist. Instead one node will ha ve the highest frequenc y in the
network. This node acts as the pacemaker and ov er time entrains the rest of the network.
Thus wa ve-lik e synchronization emer ges, where all oscillators ha ve the same frequenc y but
v arying phases φ , following:
φ ( i , t ) = φ ( d ( i , i 0 ) − ∆ φ · t ) (3.6)
where
d ( i , i 0 )
is the discrete network distance between nodes
i
and
i 0
as gi ven by the
commonly employed shortest distance algorithm by Dijkstra
208
. Node
i 0
is the fastest
node in the network and thus the source of the w a ve. Phases
φ ( t )
can be calculated from
concentration time series
c ( t )
as described in appendix A.1 .
∆ φ
is the constant phase
dif ference between neighboring oscillators with unequal distance to the source node
i 0
.
Furthermore the phase dif ference is identical to the slope of the wa ve
m = ∆ φ / ∆ d
, since the
distance measure is discrete, ∆ d = 1.
This discrete wa ve is well-established in biological settings. There it emanates from a central
pattern generator (CPG) in neural networks, which controls locomotion
209
. One neuron
after another in a chain fires and periodically initiates muscle mov ement. This leads to
swimming in case of eels or walking gaits in he xapods. The CPG architecture has also been
successfully applied to construct robots that mimic the motion of animals, such as snakes
210
or roaches 211 .
3.2 Experimental Results
In the e xperiment (Fig. 3.2 ), the network is realized with chemical micro-oscillators
212
that
measure about
300 µm
. Each oscillator consists of a porous cation-e xchange bead that is
loaded with ruthenium-tris-bipyridine (Ru(bip y)
3
), the catalyst in the oscillating Belouso v-
Zhabotinsk y (BZ) reaction. A re vie w of the chemical mechanism of the BZ reaction is giv en
in Appendix C . T o enable chemical oscillations, the particles are immersed in a catalyst-free
BZ solution. Since the concentration oscillations require the presence of the catalyst, they are
confined in space to the locations of the beads (Fig. 3.2 b). Thus in contrast to the e xperiment

32 Chapter 3
Figur e 3.2 | Setup f or network experiments with chemical micr o-oscillators.
(a) The chemical
oscillators are monitored with a camera. Based on the measured state of the oscillators they are
illuminated indi vidually with light from a spatial light modulator . For completeness, the e xperimental
setup is sho wn here, b ut is gi v en in more detail in appendix B.1.3 . (b) The oscillation cycle is
con veniently track ed optically , because the absorption spectrum of the catalyst depends on its oxidation
state. In this snapshot the beads appear red (Ru
2+
) and green (Ru
3+
) due to contrast enhancement with
the bandpass filter (
440
–
460 nm
) and a software filter . The length of the white bar corresponds to
1 cm
. (c) The network (blue arro ws) is established via light perturbations that depend on the state of
the selected neighbors.
in a spatially e xtended medium (chapter 2 ), the beads allo w for discrete reaction sites that
serv e as the oscillatory nodes in network e xperiments.
The oscillators are spaced at least
400 µm
apart. This distance suf fices to exclude dif fusi v e
coupling. Instead the coupling between oscillators is mediated via light (Fig. 3.2 c). F or this
purpose, the catalyst in the reaction has two additional roles: Its visible color changes allo w
for optically tracking the chemical oscillation cycle. Secondly , the catalyst is photosensiti ve.
Increased light application accelerates the oscillation, while a decrease slo ws the oscillation
c ycle do wn. Thus, the catalyst can be employed to realize an autonomous network e xperiment
(Fig. 3.2 a). The critical components are a camera that measures the oscillation state in the
form of grayv alues
v i
and a spatial light modulator that applies v ariable illumination lev els
I i
on indi vidual beads. Here grayv alue
v i
is short for the spatially a veraged grayv alues across
an entire bead
i
. One complication is that the oscillators are not fixated spatially , so they can
mo ve freely through the Petri dish. T o measure at the right position and tar get the correct
location with light, each oscillator is tracked continuously throughout the e xperiment.

3.2 Experimental Results 33
The e xperiment proceeds in four stages. During the initial and last stage, the oscillators are
uncoupled. All indi vidual illumination intensities are identical to the background illumination
le vel I 0 ,
I i = I 0 . (3.7)
This allo ws for the measurement of their free oscillation period
T 0
before and after the
e xperiment. In the second stage, the experiment is initialized to start from a state of in-phase
synchronization. This initial condition is enforced by e xploiting the Kuramoto transition
213
,
where globally coupled oscillators synchronize in phase for a lar ge enough coupling strength.
The corresponding illumination protocol is:
I i ( t ) = I 0 + K
N
∑
j = 1 ( v j ( t ) − v i ( t ) ) . (3.8)
Here e very component in the adjacenc y matrix
A
equals
1
(compare
( 3.3 )
). After about
5
periods, the phases of all oscillators obey the initial condition. Directly after successfully
enforcing the initial condition, the coupling stage starts. The network coupling is gi ven by:
I i ( t ) = I 0 + K
k i
N
∑
j = 1
A i j ( v j ( t ) − v i ( t ) ) . (3.9)
Each node
i
recei ves a feedback
I i
, that is based on the grayv alue differences between
neighboring nodes
j
and node
i
itself. The accumulated dif ferences are weighted with the
coupling strength
K
that is normalized by the node de gree
k i
. Real-world constraints moti v ate
the introduction of the illumination of fset
I 0
and node de gree
k i
. The intensity output range
of the spatial light modulator is bounded. Feedback v alues outside the bounds are clipped
to the minimum and maximum intensities. Clearly the minimum intensity is zero. This is
problematic in the case node
i
fires before its neighbors, because the sum in
( 3.9 )
yields a
ne gati ve v alue that would be clipped to zero. Ho we ver , with an of fset
I 0
, this case still leads
to a non-v anishing feedback. Since it is smaller than the default v alue, it will decelerate
the oscillation. In the case of an oscillator with a high degree
k i
and not too small coupling
strength
K
, it may be constantly exposed to a lar ge feedback v alue that is clipped to the
maximum intensity . Instead of being coupled to its neighboring oscillators via time-v arying
perturbations, this oscillator just recei ves an increased, time-independent intensity of fset.
Thus it is ef fectiv ely decoupled from the rest of the network. The elev ated of fset might
e ven change the dynamic beha vior of the node from oscillatory to e xcitable. Introducing
normalization by node de gree k i pre vents this scenario.

34 Chapter 3
Figur e 3.3 | W a ve synchr onization via symmetry clusters.
(a) Nodes of the network are colored
by their symmetry cluster af filiation. The numbering follo ws the path of the wa ve through the
network, starting from the pacemaker at node
i = 1
. (b) In a co-rotating phase plot with
φ − ω 1 t
,
the wa ve structure becomes apparent, as the wa ve tra vels from one shell, consisting of complete
symmetry clusters, to the next. Close to the end of the experiment, at
t = 1423 . 0 s
, the histogram of
the accumulated relati v e phases re veals stable phase-locking.
After the network is coupled together and for suf ficient coupling strengths
K
, it takes a fe w
periods for the fastest oscillator in the netw ork to entrain
38
the other oscillators. Once the
entrained nodes also oscillate with the frequency of the f astest oscillator , they in turn also
entrain their neighbors until the whole network is frequenc y-synchronized to the frequency
of the fastest oscillator (Fig. 3.3 and Fig. 3.4 ).
Nodes with the same network distance to the pacemak er node, also called shells
198
(Fig. 3.4 a),
are in-phase synchronized. This is similar to a target w a ve on a tw o-dimensional plane,
where all points with the same distance to the tar get center are in-phase synchronized as well
(Fig. 3.1 a). Regarding the symmetry properties of the netw ork, it turns out, that the wa ve
starts from one symmetry cluster and propagates further into the others. Each shell consists
of one or more complete symmetry clusters. The nodes of one symmetry cluster are not
distrib uted o ver dif ferent shells (Fig. 3.3 and Fig. 3.4 ).
The e xperimental results in figure 3.3 depict this ordered sequence. The pacemaker is labeled
as node 1. Node 1 occupies an entire symmetry cluster , colored in green (Fig. 3.3 a). The
wa ve propagates from node 1 to its ne xt neighbors which are nodes 2 and 3 as well as nodes
4 and 5, which constitute the orange and violet symmetry clusters, respecti vely . Then the
wa ve is relayed to the third and last shell, which consists of the complete magenta and c yan
symmetry clusters. This behavior repeats itself with the period of the pacemak er oscillator
T 1 = 32 . 0 s.

3.2 Experimental Results 35
Figur e 3.4 | T arget wa ve on a netw ork.
(a) The shells of the tar get wa v e on the network are
highlighted in red, blue and green. (b) In a frame co-rotating with
ω 1
the constant phase relationship
between the constituents of the shells o ver the course of the e xperiment becomes apparent. The
occurrence of each phase is proportional to its opacity . (c) Once the network coupling is acti vated
at
t = 200 s
, the shells (red, blue, green) quickly approach in-phase synchronization while the total
network (black) does not, as quantified by the K uramoto order parameter
( 3.12 )
. (d) The color in the
node-time plot sho ws the measured grayscale v alues ov er time for each node. After a transient all
nodes oscillate with the same frequency b ut a phase dif ference depending on their shell (separated
by red lines). The colorbars on the left re veal the alignment between shells and symmetry clusters.
Coupling strength: K = 2 . 0.
The phase φ is calculated from linear interpolation between consecuti ve peaks at times t i in
the grayv alue time series (Fig. 3.4 d):
φ ( t ) = 2 π t − t i
t i + 1 − t i
. (3.10)
Since the calculation of the phase
φ
during the interv al
t ∈ [ t i , t i + 1 ]
requires the time points
of the most recent peak at
t i
and the upcoming one at
t i + 1
, the phase is e v aluated after the
completion of the e xperiment. The phase dif ference between consecuti ve shells,
∆ φ = 0 . 67 π
,
is constant (Fig. 3.3 b and Fig. 3.4 b).
Figure 3.4 sho ws the node dynamics with a focus on the shells. The nodes are colored by
their distance to the pacemaker and are numbered identically as in figure 3.3 . A stationary
phase relationship between the shells is re vealed (Fig. 3.4 b) by plotting all phases of the

36 Chapter 3
constituent nodes during the e xperiment in a co-rotating frame:
φ = md + φ 0 . (3.11)
Here
d ∈ N
is the distance to the pacemaker node as well as the shell inde x. Thus the slope
m = ∆ φ / ∆ d
simplifies to the locked phase dif ference
∆ φ = 0 . 67 π
, since
∆ d = 1
between
consecuti ve shells. The parameter
φ 0
is an arbitrary phase of fset in the co-rotating frame.
This observ ed linear dependence of the phase on the distance is a strong indicator for one
dominating e xcitation wa ve spreading on the network.
Another way to reliably detect tar get wa ve induced synchronization employs the K uramoto
order parameter
213
. Once the tar get wa ve is established, for each shell, consisting of a set of
nodes S inde xed by a v ariable s , the Kuramoto order parameter ,
R S = 1
| S | ⏐ ⏐ ⏐ ⏐ ⏐
∑
s ∈ S
e i φ s ⏐ ⏐ ⏐ ⏐ ⏐
, (3.12)
should approach unity to v erify shell-wide in-phase synchronization. At the same time,
the global K uramoto order parameter
R
for all nodes must not reach one. Otherwise the
whole network is in-phase synchronized with
∆ φ = 0
between shells and does not sho w wa ve
synchronization with ∆ φ  = 0. Both conditions are fulfilled in the experiment (Fig. 3.4 c).
The space time plot analogue for a network (Fig. 3.4 d) sho ws that the wa ve pattern quickly
de velops after a short transient of fi ve periods lasting
∆ t = 150 s
and remains phase-locked
afterwards. The colorbars on the left indicate for each node the affiliated symmetry cluster
and shell. The nodes in each shell are in-phase synchronized and adjacent shells hav e constant
phase dif ferences. Also each shell consists exclusi vely of complete symmetry clusters.
The emer gence of a tar get wa ve on a network is conditional on the successful entrainment
between an y two connected nodes in the network. Ho we v er , complete entrainment is only
possible for a suf ficiently high coupling strength
K 213
. For coupling strengths
K
belo w a
critical coupling strength
K c
, wa ve propagation breaks do wn. Indeed, e xperiments with a too
small coupling strength
K < K c
confirm that the pacemaker node can not entrain the whole
network (Fig. 3.5 ). In this case nodes af filiated with the same symmetry cluster lose in-phase
synchronization and thus the network-wide tar get wa v e disappears.
While lo wering the coupling strength
K
belo w
K c
, frequenc y synchronization gradually fails.
Entrainment first breaks do wn between those pairs of connected nodes, whose frequency
dif ference
∆ ω i j
is lar ge. Furthermore the dif ference is amplified by the larger de gree of both
nodes,
max ( k i , k j ) ∆ ω i j
, due to the node degree dependence in the coupling scheme
( 3.9 )
.
This is similar to the pairwise synchronization transition of two coupled phase oscillators
213
.

3.2 Experimental Results 37
Figur e 3.5 | Entrainment failure. (a) This experiment has a coupling strength of K = 0 . 6, which is
slightly belo w the critical coupling strength guaranteeing global entrainment. Nodes 1 and 9 coexist
as pacemakers in the netw ork. Even though node 9 has the lar ger frequency , it can not entrain the
network, because the neighboring node 5 does not follo w it. The rest of the network is entrained
by the node with the second highest frequency . (b) The oscillator frequencies are di vided into two
groups with frequencies
ω 9 = 19 mHz
or
ω 1 = 17 mHz
. (c) At small coupling strength,
K = 0 . 05
,
no lasting entrainment occurs and aligned phases are only temporary . (d) The oscillators exhibit no
lasting frequency synchronization, b ut a collecti ve parameter drift to higher frequencies.
After the first of these entrainment failures occurs, the network is di vided into separate
hemispheres. Each domain is under the influence of its o wn respecti v e pacemaker , nodes 1
and 9 (Fig. 3.5 a). Interestingly , two pacemak ers coexist, e ven though their frequencies are
dif ferent,
ω 9 > ω 1
(Fig. 3.5 b). In continuous media as well as on networks, this is due to
oscillators at the boundaries of the influence sphere of each pacemaker being less e xcitable
and spending more time in their refractory states. Thus, they neither respond to nor relay the
wa ve that originates at the pacemak er .
At v ery lo w coupling strengths
K
, no pairs of connected oscillators are entrained to each
other . The space time plot of an experiment re garding this case sho ws ho w the initially
in-phase synchronized oscillators slo wly lose their phase alignment ov er time (Fig. 3.5 c).
The y do not oscillate with their natural periods, as they are still weakly coupled to each
other . Interestingly , the frequency of an oscillator is not constant o ver time. Whenev er its
phase aligns with a neighbor it slo ws do wn until the phase alignment breaks up again. So
there is temporary frequenc y-locking between oscillators, b ut it is only transient. Ov erall the
frequencies are desynchronized and o ver time increase slightly (Fig. 3.5 d) due to leakage of
the catalyst in the chemical e xperiment.
T o determine the robustness of the w a ve pattern, a lar ge number of numerical simulations
are performed with the ZBKE model of the photosensiti ve BZ reaction
( C.5 )
. The ev olution

38 Chapter 3
Figur e 3.6 | Arnold tongue f or target wa ve synchronization.
(a) Numerical simulations with
( C.5 )
for a range of coupling strengths
K
and node heterogeneities as quantified by the standard de viation
σ T 0
of the natural period distrib ution relati ve to its mean
T 0
. The y re veal the e xistence region for
synchronization by a tar get pattern. The standard de viation of the periods
σ T
during the coupled stage,
sho ws frequency-locking (purple domain) for suf ficient coupling strength regardless of heterogeneities.
The transition line is highlighted in black as a guide for the e ye. Each simulation started from a
slightly perturbed in-phase initial condition and lasted for
200
periods. (b) The time-av eraged wa ve
order parameter
( 3.14 )
re v eals that only a subset of the frequency-lock ed states is due to a target w a ve
(orange), once the heterogeneity is lar ge enough. (c) Histograms and (d) phase distrib utions marked
in subplots a,b sho w representati ve cases. Node coloring and indices are the same as in figure 3.4 .
equation for the network is:
d c i
d t = f ( c i ) + ( K u
K v ) K
k i
N
∑
j = 1
A i j ( v j − v i ) (3.13)
The two-component nonlinear node dynamics
f ( c i )
supply the oscillations. Since the light
intensity
I i
is an additi ve parameter in both components, the light-mediated coupling term
can be separated from the nonlinear kinetics
f
. Note that the dynamical equations
( 3.13 )
apart from the nonlinear dynamics f are in a similar form to the consensus protocol, which
is a well-established model of distrib uted computing in computer science
214
. The dynam-
ical v ariables
c = ( u , v )
are related to the bromous acid HBrO
2
and the oxidized catalyst

3.2 Experimental Results 39
Ru(bipy)
3+
3
concentrations. In the coupling term, only v ariable
v
plays a role, since only the
catalyst is observ able in the experiment (Fig. 3.2 ). The coupling term feeds back into both
components u and v via the v ector of species-dependent susceptibilities ( K u , K v ) .
The parameter space of the simulations is spanned by coupling strength
K
and oscillator
heterogeneity . A quantitati ve measure of the heterogeneity is gi v en by the standard de via-
tion
σ T 0
of the natural period distrib ution relati ve to the distrib ution mean
T 0
. The period
distrib ution in the simulation fits to the distrib ution in the experiment by dra wing random
samples from a suitable interv al of values for parameter q ∈ [ 0 . 5 , 0 . 9 ] .
The results of the numerical surv ey are sho wn in figure 3.6 . The standard de viation of
oscillator periods during coupling
σ T
is a good measure for e v aluating frequency synchro-
nization. A v anishing
σ T
requires the periods of all oscillators
T i
to be identical, which is
the case during frequenc y synchronization. As can be clearly seen in subfigure 3.6 a, there
e xists a transition point from a desynchronized state to a synchronized state, whose critical
coupling strength gro ws approximately linear with oscillator heterogeneity . As such this is
the generalization of an Arnold tongue
189
for a network of coupled oscillators. While this
structure is usually observ ed in the parameter space of oscillators, which are entrained to an
e xternal periodic forcing, here we hav e the case of a fast oscillator which entrains the rest of
the network as a pacemak er .
Frequenc y-locking does not of fer information about the phase alignment between the oscilla-
tors. As such a v anishing
σ T
does not dif ferentiate between global in-phase synchronization,
phase clusters
189
or tar get wa ves (Fig. 3.4 ). T o solv e this problem, a modified Kuramoto
order parameter is introduced, that detects wa v e synchronization in a network of
N
oscillators:
R wa ve = 1
N ⏐ ⏐ ⏐ ⏐ ⏐
N
∑
k = 1
e i 2 π
m φ k ⏐ ⏐ ⏐ ⏐ ⏐
. (3.14)
As in figure 3.4 b, the slope of the wa ve
m = ∆ φ
must be determined from a linear re gression
of the stationary phase pattern beforehand. Intuiti vely , this order parameter multiplies the
phase of each oscillator in such a way , that all phases con v erge to the same point if the y
are synchronized in a wa ve with slope
m
. This is possible due to the
2 π
-periodicity of the
comple x phase argument. Ho we ver , the wa ve order parameter is not v ery rob ust against noise
and gi ves ambi v alent results for too large noise le v els. W ith the additional constraint, that the
shells of the wa ve align with the symmetry clusters (Fig. 3.4 d), this order parameter can be
utilized to detect tar get wa ves on netw orks.
The simulations are all prepared in such a w ay , that the pacemak er occupies a complete
symmetry cluster (node 1 in figure 3.4 a). This guarantees, that a w a ve propag ating across the
entire network must be a tar get wa ve for suf ficiently high coupling strengths
K
. Ev aluating

40 Chapter 3
the wa ve order parameter
( 3.14 )
re veals, that tar get wa ves are the pre v alent synchronization
mode in the network of heterogeneous ZBKE oscillators (Fig. 3.6 b). A snapshot of its period
distrib ution and phase alignment is sho wn in the subplots of figure 3.6 c,d) marked with a star:
The frequencies are identical and there are three shells, which are aligned with the symmetry
clusters (compare Fig. 3.1 ).
At the edges of the frequenc y-synchronized region, wa ve synchronization f ails. For v ery
small heterogeneity , at the transition to in-phase synchronization at
σ T 0 = 0
(marked with an
upside-do wn triangle), the wa ve pattern breaks apart. As can be seen in the subplot marked
with a diamond in figure 3.6 d, there are still three separate shells as in the fully-de veloped
tar get wa ve. The difference to that case is, that f ast oscillators join a shell that is ahead of the
one the y should be in, based on the network distance (nodes 4 and 9). The opposite scenario
occurs for lar ge heterogeneity (square marker). Here oscillators fire in a later shell. The
reason for this is that the wa ve does not follo w the shortest path through the network, since
some links ha ve too lar ge a frequency heterogeneity
∆ ω i j
. Oscillators firing in the wrong
shell lead in both cases to an incorrect slope determination. This in turn results in a very
small order parameter
R wa ve
, which correctly indicates the absence of wa ve synchronization
through shells.
As observ ed in the experiment (Fig. 3.5 ), the desynchronized domain is reproduced for lo w
coupling strengths
K
. Before complete desynchronization (circular marker) there is partial
synchronization (upright triangle). Here nodes 2 and 7 oscillate with a slo wer period than the
period of the pacemaker , node 1.
The findings on tar get wa ves in netw orks are not limited to the network e xamined so
far . Figure 3.7 sho ws target w a v es for additional network topologies: random densely-
connected
202
, scale-free
215 , 216
, random sparsely-connected
217
and tree networks
218
. The
scale-free network sho wn in figure 3.7 c,d) features a lar ge number of tri vial symmetry
clusters, which consist of just a single node. Thus tar get wa ve propagation on such netw orks,
gi ven a random pacemaker location is highly probable.
Note that, if the wa ve does not originate from a complete symmetry cluster , the shells and
symmetry clusters will not align. This is the case when the pacemaker node is not a tri vial
symmetry cluster or the pacemaker site consists of equally f ast nodes that only form a subset
of a symmetry cluster . Ho we v er , in the presence of a bottleneck node, whose remo v al would
separate the network into tw o disjoint networks, w a ve propag ation through symmetry clusters
can reco ver . The necessary condition for this recov ery is, that the bottleneck node is a
tri vial symmetry cluster . In this way , once the wa ve is relayed via the bottleneck node it
acts as a pacemaker node for the remaining part of the netw ork. This scenario is depicted
in figure 3.7 e,f). Node 1 is only a subset of the magenta symmetry cluster . So, wa ve

3.2 Experimental Results 41
Figur e 3.7 | Pre valence of target wa ve synchr onization.
T arget wa ve synchronization w as observed
on (a,b) random densely-connected network with
N = 10
, (c,d) scale-free network with
N = 25
, (e,f)
random sparsely-connected network with N = 20 nodes. (g,h) tree network with N = 40 nodes. The
left column sho ws the network and its symmetry clusters. The pacemaker node is highlighted with a
black dot. On the right are space-time plots of v i ( t ) for the last 10 periods.
propagation on the left part of the netw ork is not a target w a v e. Ho wev er , once it passes the
bro wn bottleneck node in the center it is relayed symmetrically , so it re gains its tar get wa ve
character: The subsequent shells of the wa ve coincide with the symmetry clusters.
In e xperiments and simulations, it is striking that wa ve propagation follo ws the symmetry
clusters (Fig. 3.4 and Fig. 3.6 ). The reason for that is that all nodes of a symmetry cluster
ha ve the same distance to all nodes of any other symmetry cluster . This is illustrated in
figure 3.8 .
A wa ve on a netw ork spreads from its source to the neighboring nodes and in the next step
continues on to their neighbors. At each step the wa ve front increases its distance
208
to the
source node by ∆ d = 1, since the network is discrete.

42 Chapter 3
Figur e 3.8 | Relationship between symmetry clusters and shells.
(a) Any network is gi ven by a
set of nodes connected by a set of links. All nodes are assumed to be identical. (b) Every network can
be completely decomposed into symmetry clusters. (c) The quotient network is obtained by replacing
all nodes of a symmetry cluster with a single one. (d) Shells are defined by the shortest distance to
a reference node (red). Nodes with distance
d = 1
are blue, with
d = 2
are green. (e) Coloring the
nodes of the quotient network by distance to the reference node (red), re veals the same arrangement
of shells as in c).
The path of the wa ve coincides with the symmetry clusters, if the wa ve originates from all
nodes of a complete symmetry cluster . This can either be a single node of a tri vial symmetry
cluster or more (e.g. the green node or the orange nodes in figure 3.8 b). In continuing
forward the w a ve spreads to all neighboring nodes of that initial symmetry cluster . All of
these neighbors must also be complete symmetry clusters, because all nodes of the original
symmetry cluster are connected identically to the rest of the network. Otherwise the original
nodes would not constitute a complete symmetry cluster . Note that identical connecti vity is
not implied to mean that all nodes ha v e exactly the same neighbors. Instead they ha v e the
same number of neighbors from the same symmetry clusters.
The abo ve point is further clarified by eliminating the redundanc y due to the symmetry
clusters. T o this end, all nodes of a symmetry cluster are replaced by a single node with the
same connecti vity resulting in a quotient network
219
(Fig. 3.8 c). Coloring the nodes in the
quotient graph by their distance to the green node (Fig. 3.8 e), which is also the reference
node in figure 3.8 d, re v eals the same shell af filiations of nodes as in the full network. Thus all
nodes of an y symmetry clusters will fire in unison in the case of tar get wa ve synchronization.

3.3 Short summary 43
3.3 Short summary
This chapter generalizes the notion of tar get wa ves from continuous media to discrete
networks utilizing symmetries. T arget w a ves on the tw o-dimensional Euclidean plane
spread while adhering to all members of the Euclidean symmetry group E(2): translation,
rotation and reflection. Lik e wise tar get wa ves on netw orks obey all permutation symmetries
underlying the network topology . The source of the tar get wa ve is the pacemak er , a location
or node, which features the lar gest frequency in the entire system. After a suf ficiently long
time, each element is entrained to the frequenc y ω 1 of the pacemaker .
T arget w a v es on networks are studied in an e xperiment with discrete photochemical oscillators
(Fig. 3.4 ) and reproduced in numerical simulations (Fig. 3.6 )
217
. For this purpose, a w a ve
order parameter was de veloped
( 3.12 )
to reliably detect and distinguish tar get wa ves from
other synchronization modes. Experimentally and numerically the transition from netw orks,
which are frequenc y-locked by the tar get wa ve, to desynchronization is v erified to occur by
entrainment failure along connected nodes
i , j
whose frequenc y dif ference
∆ ω i j
is too lar ge.
An analysis of the network symmetries further re veals their intrinsic connection to the netw ork
distance. This allows for the prediction of w av e propagation using symmetry clusters.
The e xistence of tra velling cortical e xcitation wa ves w as recently observ ed in the motor
corte x of monke ys (Macaca mulatta)
220
. Here, the findings with neuromorphic chemical
oscillators re veal that tar get wa ve-induced synchronization may be responsible for rob ust in-
phase synchronization of distant nodes in a cortical network without a delayed transmission
node 180 .
3.4 Futur e dir ections
W ith the experimental setup presented in this chapter it w as only possible to study the
transition from desynchronization to tar get wa ves at lar ge heterogeneity . T o study the
transition from in-phase to wa ve synchronization (Fig. 3.6 ) experimentally , it is necessary
to ha ve precise control o ver the period distrib ution. For this purpose it is required to ha ve
lar ge reserv oir of
N > 2000
oscillators, which are a v ailable in the setup described in the next
chapter . Before the experiment runs, it is possible to select
N = 10
oscillators with a desired
period distrib ution. Furthermore, the lar ge reserv oir allo ws for more than 200 experiments to
run in parallel. Thus a surv ey of the synchronization modes in the
K
-
σ T 0
parameter space
can be completed quickly .
It would be interesting to emplo y weakly coupled oscillators as a first approximation of the
critical coupling strength
K c ( σ T 0 )
at which a network transitions from the desynchronized to

44 Chapter 3
the synchronized state (Fig. 3.6 ). In the case of two mutually coupled oscillators, the critical
coupling strength K c at which they frequenc y-lock is gi v en by 213
K c = ∆ ω
max IF ( ∆ φ ) . (3.15)
The critical coupling strength depends on the frequency dif ference
∆ ω = ω 1 − ω 2
between
both oscillators and the maximum of the interaction function
IF ( ∆ φ )
, which is defined as the
odd part of the phase response curv e of an oscillator (see appendix A.1 ).
The case of a pacemaker sequentially entraining one oscillator after another , can be simplified
to two oscillators. One is entrained by the pacemaker and oscillates with its frequenc y
ω 1
.
The other is not entrained yet and has a frequency
ω i
. Thus, accounting for the degree
normalization in
( 3.9 )
, the critical coupling strength
K c
at which e very oscillator in the
network is entrained follo ws
K c = ∆ ω max
max ( IF ( ∆ φ )) , with (3.16)
∆ ω max = max
i ( k i ( ω 1 − ω i )) . (3.17)
Ev en though, this approximation neglects the influence of neighboring oscillators
j
, which
would alter
ω i
, it might still yield reasonable results. Ho wev er , its applicability to ZBKE
oscillators might be limited, since their phase response curve depends nonlinearly on the
coupling strength (see appendix C.2 ).

Chapter 4
Spiral W a v e Chimera

46 Chapter 4
A spiral wa ve chimera is the union of spiral w a ves
87
and chimeras states
221
- two paradigms
in spatial pattern formation and temporal synchronization
222
. Spiral wa ves ha ve been
researched e xtensi vely in simulations as well as e xperiments during the last 70 years
62
in
e xcitable media due to their spontaneous formation in a plethora of natural systems (see
the introduction of chapter 2 for e xamples). A spiral wa ve nucleates from the open end of
an e xcitation wa ve. The open end curls in and becomes the center of the spiral wa ve from
which wa ves are periodically emitted. The chimera state was numerically found by Y oshiki
K uramoto about 15 years ago
a
, when he extended his model for synchronization in netw orks
from globally to nonlocally coupled oscillators. While dissipati ve oscillators with identical
frequencies in a globally coupled system tri vially synchronize, this is not the case for nonlocal
coupling. T wo groups emer ge: One coherent group, which is frequency-synchronized and
another incoherent one, which is desynchronized.
Chimera states are theorized to play a role in cardiac tissue, where mechanical contraction
leads to an ef fectiv e nonlocal coupling
225
, opinion formation in social groups
226
, beam
shaping in laser arrays
227
, computers based on arrays of optomechanical resonators
228 – 230
,
blackouts in po wer grids
231
, SQUID metamaterials
232 – 234
and arrays of spin-torque oscilla-
tors for miniaturized antennae
235 , 236
, quantum systems of ultracold atoms in a Bose-Einstein
condensate
237
, cortical networks, where they might play a role in b ump states
238 , 239
and
epileptic seizures
240 – 242
, as well as hydrodynamically coupled cilia carpets
88
. The latter
control for e xample the flo w of nutrients in the brain to sustain neuronal acti vity 243 .
Recently chimera states ha ve been found in a v ariety of chemical, physical and biological
e xperiments with: coupled photochemical
244 , 245
and electrochemical
246
oscillators, photo-
electrodissolution of a doped silicon layer
247 – 249
, mercury-beating heart
250
, electro-optical
coupled map lattices
251 , 252
, mechanical pendula
253 – 255
, time-delayed lasers
256 , 257
, field-
programmable gate arrays
258
, electronic circuits
259 , 260
and cilia chains
261
. Even before
there was an e xperimental focus on finding chimera states, there were already experiments
which e xhibited coexistence of coherent and incoherent beha vior , e.g. in a globally coupled
photosensiti ve Belouso v-Zhabotinsky (BZ) medi um
262
and in ferrofluidics
263
. Ho wev er , the
spiral wa ve chimera remained elusi ve so f ar .
In this chapter , I will present the e xperimental verification of a spiral w a ve chimera in
a network of photocoupled chemical oscillators. Be yond the observ ation, a number of
unanticipated beha viors are disco vered as well.
a
In 2001 Kuramoto presented his findings on nonlocally coupled systems, that already encompassed one-
and two-dimensional problems, at a meeting named "Nonlinear Dynamics and Chaos: Where do we go from
here?" in Bristol, United Kingdom. Subsequently his work, on what later became kno wn as chimera state
223
,
was published as a chapter 41 in the accompanying conference monograph 224 .

4.1 Theoretical Background 47
4.1 Theor etical Backgr ound
Inspired by bacterial biofilms
264
as well as li ving cells in a body
87
, that instead of communi-
cating directly , do so via exchange of biochemical species with their surroundings, K uramoto
analyzed the follo wing model 41 :
d u
d t = f 1 ( u , v ) + K ( w − u ) , (4.1)
d v
d t = f 2 ( u , v ) , (4.2)
d w
d t = 1
ε ( u − w + D w ∆ w ) . (4.3)
The first two equations form a subsystem that exhibits a stable limit c ycle. The third
v ariable
w
is the only species that may dif fuse with diffusion coef ficient
D w
and couples
linearly into
u ( 4.2 )
. Adiabatically eliminating
( ε → 0 )
the dynamics of the dif fusing
v ariable ( 4.3 ) leads to an inhomogeneous ordinary differential equation in space for w ,
D w ∆ w − w = u (4.4)
that may be solv ed with the Green’ s function method
265
. The solution is a nonlocal inte gral
operator ,
w ( r , t ) =
+ ∞
∫
− ∞
d 2 ˜ r G ( | r − ˜ r | ) u ( ˜ r , t ) , (4.5)
whose e xact form depends on the topology of the system
41 , 266
. For a one-dimensional ring,
the kernel G is a decaying e xponential 41 ,
G ( r ) = 1
2 κ e − r / κ , (4.6)
while for an infinite two-dimensional plane with Neumann boundaries, it is the zeroth
modified Bessel function of the second kind K 0 267 ,
G ( r ) = 1
2 π κ 2 K 0 ( r / κ ) , (4.7)
whose beha vior resembles an e xponential decay similar to
( 4.6 )
. The characteristic coupling
length
κ
depends on the dif fusion coef ficient,
κ = √ D w
. Inserting the nonlocal operator
( 4.5 )
with the Green’ s function
( 4.7 )
into the original reaction-dif fusion equation ( 4.1 - 4.2 ), leads

48 Chapter 4
to an equation for oscillators with nonlocal coupling 266 , 267 :
d u
d t = f 1 ( u , v ) + K
+ ∞
∫
− ∞
d 2 ˜ r 1
2 π κ 2 K 0 ( | r − ˜ r |
κ ) ( u ( ˜ r ) − u ( r ) ) , (4.8)
d v
d t = f 2 ( u , v ) . (4.9)
Depending on the interaction length
κ
, equation 4.8 can describe systems ranging from local
to global coupling. This point becomes clear
268
by considering the nonlocal coupling in a
one-dimensional system:
d u
d t = f 1 ( u , v ) + K
2 κ
+ ∞
∫
− ∞
d ˜ x e xp ( | x − ˜ x |
κ ) ( u ( ˜ x ) − u ( x ) ) . (4.10)
T o see ho w local coupling arises in the case of
κ → 0
, we e xpand
u ( ˜ x )
in the neighborhood
of x as a T aylor series,
u ( ˜ x ) = u ( x ) +
∞
∑
n = 1
1
n !
∂ n u
∂ x n ⏐ ⏐ ⏐ ⏐ ⏐ ˜ x = x
( ˜ x − x ) n , (4.11)
and plug it into
( 4.10 )
. While the zeroth order cancels, the remaining terms lead to an integral
of the form
1
n !
+ ∞
∫
− ∞
d y y n e xp ( − | y |
κ ) = ⎧
⎨
⎩
κ n + 1
n ! n e ven
0 n odd . (4.12)
Since
κ
is small, it is possible to truncate the series after the second order . This approximation
re veals local dif fusi ve coupling:
d u
d t = f 1 ( u , v ) + K κ 2 ∂ 2 u
∂ x 2 . (4.13)
Con versely the kernel function assumes identity as the coupling range gro ws,
G κ → ∞
− − − → 1
.
Since each element is weighted identically , the coupling ( 4.10 ) is global:
d u
d t = f 1 ( u , v ) + lim
κ → ∞
K
2 κ
+ ∞
∫
− ∞
d ˜ x ( u ( ˜ x ) − u ( x ) ) . (4.14)

4.1 Theoretical Background 49
In this case,
κ
acts as a normalization constant for the di ver ging inte gral, such that the
coupling term results in a finite v alue.
Remarkably , nonlocal coupling with finite coupling range
κ < ∞ ( 4.10 )
is well approximated
by local dif fusion
( 4.13 )
. The approximation is v alid for small
κ
or equi valently for systems,
which e xhibit patterns that v ary on a length scale
λ p
being much lar ger than the coupling
range 41 :
λ p ≫ κ . (4.15)
Put dif ferently , the pattern is spatially smooth. The length scale
λ p
of a smooth pattern
described by dif fusiv e coupling ( 4.13 ) is related to the ef fecti ve dif fusion coefficient:
λ p ∼ √ K κ . (4.16)
Combining the preceeding ar guments
( 4.15 )
and
( 4.16 )
leads to a consistenc y condition for
the local-coupling approximation:
√ K κ ≫ κ . (4.17)
Thus, for lar ge coupling strength
K
the approximation is v alid. Both equations,
( 4.10 )
and
( 4.13 )
, describe smooth patterns. Ho we ver , for suf ficiently small
K
the peculiar ef fects
of nonlocal coupling dominate as non-smooth patterns emer ge. These patterns break the
dif feomorphism
269
between physical and phase space. Intuiti vely this means that tw o
infinitesimally close points in physical space e xhibit states, which are not close in phase
space 41 .
An intuiti ve e xplanation of the underlying mechanism generating non-smooth, discontinuous
patterns is re vealed during the analysis of nonlocally coupled oscillators. The limit cycle
dynamics of equations
( 4.1 )
and
( 4.2 )
can be simplified by introducing a scalar phase
v ariable ϕ via a phase reduction technique 213 (see appendix A.1 ):
d ϕ
d t = ω + K
2 κ
+ ∞
∫
− ∞
d ˜ x e xp ( | x − ˜ x |
κ ) sin ( ϕ ( ˜ x ) − ϕ ( x ) − α ) (4.18)
Due to the phase reduction, the magnitude of the nonlocal coupling
( 4.18 )
does not depend
an ymore on the dif ference of v ariables
( 4.10 )
, but on the model-dependent
2 π
-periodic
interaction function. For weakly coupled prototypical Stuart Landau
266
and FitzHugh-
Nagumo
267
oscillators the interaction function closely resembles a simple harmonic. T aking

50 Chapter 4
one further step, Kuramoto’ s mean field approach
213
for studying the synchronization of
globally coupled networks of oscillators
( A.32 )
can be e xtended to the nonlocal case. As a
result, the corresponding single-oscillator equation, which describes the interaction between
a single oscillator and the mean field via its modulus
R
and phase
Ψ
, becomes spatially
dependent 266 , 267 :
d ϕ ( x , t )
d t = ω + K R ( x ) sin ( Ψ ( x , t ) − ϕ ( r , t ) − α ) . (4.19)
This sho ws that in the nonlocal case, coupling strength
K
is modulated by the local coherence.
The le vel of coherence is quantified by the spatially v arying modulus
R ( x )
of the order
parameter . This modulus is equi v alent to a localized Kuramoto order parameter , which can
be approximated by
R ( x ) = 1
| Ω ( x ) | ⏐ ⏐ ⏐ ⏐ ∫ Ω ( x )
d ˜ x e i ϕ ( ˜ x ) ⏐ ⏐ ⏐ ⏐
, (4.20)
measuring the le vel of in-phase synchronization
189
in a nonlocal neighborhood of position
x
Ω ( x ) = { x , ˜ x ∈ R ⏐ ⏐ | x − ˜ x | ≤ ℓ R } . (4.21)
The parameter
ℓ R ∈ R +
is chosen as to co ver a suf ficiently small neighborhood, such that the
localized order parameter is not smoothed out. If all oscillators mo v e in unison, the ef fecti ve
coupling strength
K R ( r )
will be lar ge. In turn this will recruit further neighboring oscillators
for near zero-lag synchronization. Ho wev er , in an area with greatly v arying phases, the
coupling K R ( r ) v anishes, leading to e v en lo wer synchronization le vels (Fig. 4.1 ).
This also e xplains the rob ust structure of the spiral wa ve chimera. In the wa ve that rotates
around the core, there is only a small and smooth phase gradient. This leads to a lar ge
order parameter and re-enforces the high synchronization le vel. In the core region, all
phases ranging from
0
to
2 π
occur simultaneously due to the phase singularity
b
. The
order parameter nearly v anishes here and the resulting coupling is so weak, that frequency-
entrainment fails. Thus, a spiral-shaped phase pattern naturally gi v es rise to incoherent region
under nonlocal coupling. Consequently , there is no sensiti ve dependence on initial conditions,
as is the case for the chimera state on a network with tw o subpopulations 270 or a ring 271 .
In one dimension the mechanism is qualitati vely dif ferent: Attracti ve dif ference coupling
( 4.18 )
alw ays leads to in-phase synchronization as it remov es phase dif ferences (appendix A.1 ). A
b
Since the pattern is discontinuous, the phase singularity is not localized at a single point, as for re gular
spiral wa ves with a continuous concentration field. Instead the singularity is spread out ov er the entire core
region.

4.1 Theoretical Background 51
Figur e 4.1 | Order in the phase field of a spiral wa ve.
(a) A snapshot of the phase field for a rotating
spiral wa ve. In the center is the phase singularity , where all phases
ϕ
ranging from
0
to
2 π
meet. The
dashed square with sidelength
2 ℓ R + 1
and
ℓ R = 2
is the nonlocal neighborhood of an oscillator close
to the phase singularity ov er which the localized Kuramoto order parameter
( 4.20 )
is e v aluated. (b)
The resultant localized Kuramoto order parameter sho ws lo w order
( R < 0 . 2 )
inside the core and high
order ( R > 0 . 6 ) outside.
phase singularity , which might counteract this tendency , requires two spatial dimensions, so
it does not occur on a one-dimensional ring. In order to oppose e ventual in-phase synchro-
nization the phase-frustration parameter
α
in the coupling function is utilized. Exploiting
trigonometric identities, the interaction function decomposes into tw o antagonistic parts, that
promote ( sin ( ∆ ϕ )) and oppose ( cos ( ∆ ϕ )) in-phase synchronization:
sin ( ∆ ϕ − α ) = cos ( α ) sin ( ∆ ϕ ) − sin ( α ) cos ( ∆ ϕ ) (4.22)
≈ ε sin ( ∆ ϕ ) − cos ( ∆ ϕ ) . (4.23)
Approximating both parts
( 4.22 )
for
α = π / 2 − ε
, with
ε
small, leads to
( 4.23 )
. In this
range chimera patterns are likely to be observ ed
41
. Comparing the coef ficients of
( 4.23 )
,
re veals that in-phase synchronization
( ∆ ϕ = 0 )
is hea vily penalized
( 1 )
and only weakly
encouraged
( ε )
. In summary the desynchronized domains of the one-dimensional chimera
are re-enforced not due to a topological cause, as in two dimensions, b ut via suppression of
in-phase synchronization in the coupling.
Due to lar ge theoretical interest ov er the last years in states of partial synchronization in
general and chimera states in particular , there is no w a large body of research. Chimera
states were observ ed numerically in one
271 – 276
, two
60 , 271 , 277 – 281
and three
282 – 284
dimensions
as well as dif ferent network topologies
285 – 289
. They occur in a v ariety of discrete and
continuous dynamical systems exhibiting bistability
290
, oscillations
41 , 291
, e xcitability
292

52 Chapter 4
and chaos
293 , 294
. In addition they also persist under detrimental influences such as time
delay
295 – 297
, noise
298 – 300
and time-v arying network topologies
301
. Be yond that the position
and lifetime of chimera states are amenable to control
302 – 305
. Consequently , the gro wing
v ariety of chimera states prompted the dev elopment of a classification scheme
306
. A more
detailed summary of endea v ors in this direction also recently appeared
221
. The e xperimental
v erification of these findings is often limited by the large number of oscillators required.
In re gards to spiral wa ve chimeras, their e xistence and stability properties in a continuous
system were analyzed by Carlo Laing
307 , 308
. Utilizing the Ott-Antonsen ansatz
309
allo ws for
calculating the e volution equation of the order parameter
R ( x )
in the continuum limit, where
it was found that spiral w a ve chimeras persisted. Thus, the y are not artifacts of numerical
discretization schemes.
Furthermore, Ste ven Strogatz and co work ers were able to calculate the size of the incoherent
core as well as the rotation frequency of the rotating w a v e in the case of Kuramoto phase
oscillators. They combined the kno wn spiral phase field description
310
and simplified the
nonlocal kernel operator to a Gaussian, which allowed for the analytic treatment of the
coupling inte gral 311 . They concluded their article by posing the follo wing challenge:
The possibility of observing spiral wa ve chimeras in physical systems natu-
rally arises. [...] W e lea v e the experimental observ ation of chimera states as a
challenge to others.

4.2 Experimental Setup 53
4.2 Experimental Setup
Figur e 4.2 | Experimental Setup.
(a) A reserv oir of chemical micro-oscillators is fixated on an
acrylic glass plate in a thermostatted reactor
126
. During their oscillation cycles the y emit fluorescence
photons, which are recorded with a CMOS camera as grayv alues
v i
of each oscillator
i
. The v alues are
sent to a computer to determine the projected light intensity
I i
according to
( 4.25 )
. This feedback is
applied to the oscillators with a projector . (b) A camera snapshot of the reserv oir of
2816
oscillators.
Each white dot corresponds to a single chemical oscillator . The black bar represents
1 cm
and the
image is to scale at 1:2.2 (c) Spectral observ ation: The projector emits a spatiotemporal pattern
at a wa velength of
440 nm
(filled blue curve); the light is absorbed (blue curv e) and e xcites the
photosensiti v e catalyst, which leads to the emission of fluorescence light abov e
550 nm
(red filled
curve). The light from the projector is filtered by a bandpass filter with a cut-of f wa velength at
500 nm
(yello w line). Further details on the experimental setup can be found in appendix B.1.5 .
The dif ficulties associated with the experimental v erification of the spiral wa ve chimera and
ho w they are resolv ed is the content of this section. The main challenge is to ov ercome the
oscillator number limit. In addition it is required, that each oscillator can be controlled and

54 Chapter 4
Figur e 4.3 | Fixation of chemical micro-oscillator .
(a) A top-vie w photograph of an early prototype
of the acrylic glas plate with drilled wells. Each cylindrical well holds a spherical bead, that is loaded
with catalyst. The white bar corresponds to
200 µm
. (b) A schematic side vie w of chemical beads
during the experiment. The BZ reagents required for the oscillatory reaction are exchanged through
the hydrogel layer (indicated by black arro ws).
monitored indi vidually . Setups used for pre vious experiments on synchronization focused
on small populations
217 , 228 , 244 , 253 , 258 , 312 – 316
. Scaling up the size of the network be yond
N > 1000
oscillators, is often unfeasible. Ho we ver , numerical simulations indicate (see ne xt
section 4.3 ), that an array of oscillators spanning
N = 32 × 32 = 1024
nodes is required to
resolv e the incoherent core and coherent rotating arm.
In principle, the catalyst-loaded cation-exchange beads pre viously utilized for studying
synchronization on small networks (chapter 3 ) are a good candidate. The problem with
the nai ve approach of simply increasing their number from
N = 10
to
N > 1000
in a Petri
dish (see Fig. 3.2 ) is that each bead is mobile. T racking each oscillator location is not a
remedy an ymore, since ov er time all beads will stick to each other after colliding and form a
continuous medium 317 .
The solution is to place each bead within the confines of a small ca vity (Figs. 4.2 and 4.3 ).
F or this purpose we utilize an acrylic glas plate that has
64 × 44
c ylindrical wells drilled into
it (Fig. 4.2 b). All wells ha ve a depth of
150 µm
, a diameter of
200 µm
and are separated from
their respecti ve ne xt neighbors by
400 µm
. Each well holds a cation-exchange bead that is
loaded with the photosensiti ve catalyst of the BZ-reaction (appendix C.1.3 ). Furthermore, the
wells are sealed of f with a
200 µm
high layer of silica hydrogel (Fig. 4.3 b). Since the hydrogel
is microporous
318
, it allo ws for the passage of BZ reagents, such as hydrogen ions H
+
,
bromate BrO
–
3
and malonic acid MA. They are required for the oscillatory reaction to occur

4.2 Experimental Setup 55
at the bead sites. In addition, we employ a v ariant of the photosensiti ve catalyst, Ru(dmbipy)
3
,
with additional dimethylene ligands
126
. This sterically fixates the catalyst molecules inside
the bead polymer matrix, such that the y can not escape. As a consequence, chemical aging
ef fects are greatly reduced, which extends the potential runtime of e xperiments to beyond 24
hours.
Another challenge is measuring the current state of the oscillation c ycle. The traditional
method of absorption spectrophotometry
144
, where the optical contrast originates from the
dif ference in absorption spectra at v arying oxidation states of the catalyst (chapter 2.2 ), is not
applicable here. There is only a vanishingly small contrast, because the amount of catalyst
on each oscillator is too miniscule. While the lacking contrast could be improv ed using
specialized image software, there is a better alternati v e. Instead of relying on absorption,
a highly enhanced optical contrast is obtained e xploiting the fluorescence of the catalyst
(Fig. 4.2 c). In the reduced form, Ru(dmbipy)
2+
3
, the catalyst emits fluorescence photons
with a wa velength of
λ > 550 nm
whereas in the oxidized form, Ru(dmbipy)
3+
3
, it does not.
During the oscillation c ycle, the catalyst will periodically switch between both oxidation
states. This allows for the direct observ ation of chemical oscillations with a grayscale camera
that records the fluorescence photons with spatial resolution.
Due to its photosensiti vity , the catalyst can not only be e xploited for observ ation of the oscil-
lation c ycle, but also for its perturbation. Additional light intensity accelerates the oscillation,
while less decelerates it. These are the prerequisites for e xperiments on synchronization.
While the simplest cases of frequenc y-locking in periodic forcing and mutual coupling can
be successfully reproduced, the e xperimental possibilities are far from e xhausted. The light
interaction opens the possibility for a v ery general coupling scheme in v olving
N
oscillators:
I i ( t ) = I 0 +
N
∑
j = 1 ( W i j ( t ) H i j ( v i ( t , τ ) , v j ( t , τ ) , t ) ) + D i ( t ) ξ i . (4.24)
There is a background intensity
I 0
, which enables cumulated perturbations from the network
that reduce the applied light intensity . The topology of the system is defined by the weighted
adjacenc y matrix
W
. In addition to encoding the connecti vity of the network, it also holds
information on the strength of each link. For identical coupling strength
K
across all links,
W
simplifies to
W = K A
, where
A
is the adjacenc y matrix. Possible topologies include
one-, two- and three-dimensional grid networks as well as prototypical and real-w orld
networks, such as scale-free
216
, small-world
319
, sparsely and densely random networks with
symmetries
202 , 217
as well as po wer grids
320
and the complete connectome of the nematode C.
Ele gans
321 , 322
. The coupling function
H i j
may be dif ferent for each edge
i → j
, b ut alw ays
depends on the measured grayv alues
v i
at a current
v i ( t )
or past time
v j ( t − τ )
. Furthermore

56 Chapter 4
the coupling strength and type are not stationary , but can be time-dependent and thus allo w
for the implementation of control schemes. Last but not least it is also possible to include
additi ve noise, where each node has an indi vidual noise intensity
D i
and white or colored
noise spectrum 323 , 324 determining ξ i .
The general coupling
( 4.24 )
includes as a special case the nonlocal coupling
( 4.8 )
required
for the v erification of spiral wa v e chimeras. In a concise notation, the nonlocal coupling on a
two-dimensional grid netw ork is gi ven by
I i ( t ) = I 0 + K ∑
j ∈ Ω i
e − r ( i , j ) / κ ( v j ( t − τ ) − v i ( t ) ) , (4.25)
with a characteristic coupling range κ , a v ector -v alued node index,
i = ( i x , i y ) , (4.26)
that accounts for two dimensions and a corresponding Euclidean distance function
r ( i , j ) = ∥ i − j ∥ 2 = √ ( i x − j x ) 2 + ( i y − j y ) 2 . (4.27)
The square area
Ω i
on which the discrete coupling inte gral is e v aluated stems from the
maximum norm
Ω i = { i , j ∈ Z 2 ⏐ ⏐ ⏐ ∥ j − i ∥ max ≤ ℓ } = [ i x − ℓ, i x + ℓ ] × [ i y − ℓ, i y + ℓ ] , (4.28)
where
2 ℓ + 1
is the side length of the square as in
( 4.21 )
. Furthermore a v ariety of initial
conditions are accessible, because each oscillator can be manipulated indi vidually . Employing
periodic forcing close to the unperturbed oscillation period
T forcing ≈ T 0 , i
, it is possible to
entrain a node i , such that it oscillates according to
d ϕ i
d t = ω forcing + ϕ 0 , i (4.29)
with an oscillator -dependent phase of fset
ϕ 0 , i
. This allo ws for global in-phase synchroniza-
tion, phase gradients, chimera states and more as initial conditions using light.
Still, the photosensiti vity brings to the forefront yet another predicament: The heterogeneity
of the oscillators. Preliminary experiments with coupled oscillators re vealed, that the distri-
b ution of unperturbed periods
T 0 , i
was too broad, as to allo w for frequency synchronization.
W ithout it, chimera states are unobservable, because the y coexist with the globally in-phase
synchronized state
270
. A measurement of the bead diameters re veals that the bead sizes

4.2 Experimental Setup 57
Figur e 4.4 | Oscillator heterogeneity .
(a) Comparison of oscillator bead diameters before (gray) and
after sie ving procedure (green). (b) Comparison of catalyst loading for manual mixing (gray) and by a
v ortex mix er (green). In both cases the occurrence is normalized to the initial size of the distrib ution.
are not monodisperse (Fig. 4.4 a). The bead diameters approximately follo w a Gaussian
distrib ution ranging from
75
–
150 µm
centered at
110 µm
. Pre viously the size distrib ution
of beads was identified as a source of period heterogeneity
325
. The reason is that larger
beads feature a lar ger surface area allo wing for a greater reactant flux. This influences the
resulting periods. In addition each bead – e ven of identical size – may hold dif ferent amounts
of catalyst (Fig. 4.4 b).
W ith the root causes of the period heterogeneity identified, it is possible to rectify them.
T o homogenize the size distribution, we built our o wn sie ving machine. It consists of a
frequenc y generator , a car amplifier stage, a subwoofer speaker , different fine sie ves and
tape. A detailed description of the sie ve machine is in the appendix B.1.5 . The mesh sizes
of the three stacked sie ves are:
106 µm
,
112 µm
and
125 µm
. The bead throughput is greatly
enhanced when adding tin y glas spheres with a diameter of
1 mm
to each sie ve, because their
kinetic ener gy pushes the beads through the holes. After the sie ving procedure is finished,
the remaining beads are collected between sie ves with meshes 106 µm and 112 µm.
Homogeneous catalyst loading of these beads is achie ved by slo wly adding the catalyst to a
vial of beads in water solution. It is critical that this is done while the vial is continuously
shaken by a v ortex mix er .
The final result of this procedure is quantified under a light microscope. Utilizing image
recognition software to detect the beads as circles on a camera snapshot, it is possible to
subsequently e xtract their diameters
d
and a v erage color saturation
c
(Fig. 4.4 ). In this way
their quality is automatically and quickly assessed. This re veals, that the beads ha ve a v ery
narro w size distribution (100 –120 µm) and are loaded nearly identically with catalyst, such
that the y are suitable for experiments on spiral w a ve chimeras.

58 Chapter 4
T aken together each immobile bead is a node in the reserv oir of chemical oscillators. Depend-
ing on properties such as their natural period
T 0
and their phase response curv e, all beads are
filtered. Suitable candidates are utilized as nodes in a network e xperiment that is connected
to its neighbors via light. Exploiting the fluorescence properties of the catalyst allows for a
simple yet compact e xperimental setup (Fig. 4.2 ).
4.3 Numerical Simulations
In order to guide the search for the spiral wa ve chimera in the e xperiments, a large number
of numerical simulations were performed to e xplore the space of coupling parameters.
The dynamics of the catalyst loaded beads can be reproduced qualitati vely with the ZBKE
model
326
. It was originally de vised by Zhabotinsk y and Epstein in 1993 and later en-
hanced
244 , 262 , 327
to account for the e xcitatory ef fect of light illumination
328
(see appendix C.1.3
for a brief re view of the deri v ation). Here, the model is adapted to describe the local dynamics
of an oscillator i = ( i x , i y ) in a network:
d u i
d t = 1
ε 1 ( I i + µ − u i
µ + u i ( β + q i
α v i
ε 3 + 1 − v i ) + γ ε 2 w 2
ss , i + ( 1 − v i ) w ss , i − u 2
i − u i ) ,
d v i
d t = 2 I i + ( 1 − v i ) w ss , i − α v i
ε 3 + 1 − v i
,
w ss , i ( u i , v i ) = 1
4 γ ε 2 ( √ 16 γ u i ε 2 + v 2
i − 2 v i + 1 + v i − 1 ) .
(4.30)
In this model the v ariables
u i
,
v i
and
w ss , i
represent the dimensionless concentrations of
bromous acid [HBrO
2
], oxidized catalyst [Ru(dmbipy)
3+
3
] and bromous acid radical in
equilibrium [HBrO
+
2
]
ss
, respecti vely . Chemical model parameters and their v alues are listed
in table 4.1 together with parameters of the coupling.
T wo parameters stand out due to their special role: The period heterogeneity is introduced
heuristically
244
by dra wing the stoichiometric parameter
q i
of each bead from a uniform
distrib ution
q i ∈ [ 0 . 5 , 1 . 0 ]
, which leads to a distrib ution of natural periods
T 0 ( q i )
between
30 . 2
and
45 . 9
. In relation to the reference period
T 0 = 34 . 4
at
q = 0 . 7
, the limits are
0 . 88 T 0
and
1 . 33 T 0
. Note that, since the model is dimensionless, absolute time durations are giv en
without units. The resulting spread of periods in the simulations agrees well with the
e xperiments. Parameter
I i
represents the light intensity applied on oscillator
i
. This is the
most important parameter , because it plays the central role of introducing nonlocal coupling.

4.3 Numerical Simulations 59
In the simulations the oscillators on a two-dimensional grid of
n x × n y
nodes are coupled
according to
I i ( t ) = I 0 + K ∑
j ∈ Ω i
e − r ( i , j ) / κ ( v j ( t − τ ) − v i ( t ) ) . (4.31)
This is the same coupling formula as used earlier for the experiments
( 4.25 )
, since the light
interaction introduced via
I i
is additi ve in the local dynamics
( 4.30 )
. The time-delayed dif fer-
ences in grayv alues between the oscillator
i
and its nonlocal neighbor
j
are weighted with
an e xponential kernel, that decreases with the Euclidean distance
r ( 4.27 )
. The parameter
κ
is the characteristic coupling length. For small
κ
the coupling is v ery localized, while for
lar ge
κ
it encompasses distant nonlocal neighbors. The neighbors are taken from a square
re gion Ω i with side length 2 ℓ + 1 centered on oscillator i ( 4.28 ).
T able 4.1 | ZBKE model.
For a deri v ation of the v alues based on the reagent concentrations in the
experiment, see appendix C.2 . Note that the stif f dynamics of the ZBKE model require the use of the
double precision datatype. Simulations with single precision floating point v alues de velop numerical
artifacts, that lead to strong de viations from the original limit cycle.
v ariable / parameter v alue description
u i ( t ) dimensionless [HBrO 2 ] on node i
v i ( t ) dimensionless [Ru(dmbipy) 3+
3 ] on node i
w ss , i ( u i , v i ) dimensionless steady state [HBrO +
2 ] on node i
i = ( i x , i y ) two-dimensional inde x
I i light intensity projected on node i
τ time delay
K 5 . 25 × 10 − 4 coupling strength
κ 2 . 5 coupling range parameter
ℓ 4 maximum coupling distance
∆ t 1 . 0 × 10 − 4 integration time step
I 0 5 . 25 × 10 − 4 background light intensity
q i 0 . 5-1 . 0 stoichiometric parameter of node i
ε 1 0 . 11 ⎫
⎬
⎭
time scale parameters
ε 2 1 . 7 × 10 − 5
ε 3 1 . 6 × 10 − 3
α 0 . 1 ⎫
⎪
⎪
⎬
⎪
⎪
⎭
kinetic parameters
β 1 . 7 × 10 − 5
γ 1 . 2
µ 2 . 4 × 10 − 4

60 Chapter 4
Dif ferent types of nonlocal kernels, such as a constant
277
or Gaussian
311
did not sho w
qualitati vely dif ferent results. For simplicity the employed k ernel is not normalized as in
pre vious work
267
. Here, a possible normalization factor just rescales the coupling strength
K
.
Since the grid is finite, it is important to account for boundary ef fects. Generalizing the
nonlocal coupling in
( 4.31 )
by replacing the time-delayed grayv alue dif ference with a
coupling function H ( v i , v j ) leads to
I i ( t ) = I 0 + K
N
∑
j ∈ Ω i
e − r ( i , j ) / κ H ( v i , v j ) . (4.32)
At the boundary , the general coupling function H is modified as,
H ( v i , v j ) = ⎧
⎨
⎩
v j ( t − τ ) − v i ( t ) , 1 ≤ j x ≤ n x ∧ 1 ≤ j y ≤ n y
0 , else , (4.33)
in order to omit non-e xistent nodes beyond the grid. This procedure is similar to Neumann
boundary conditions 158 . Whereas for periodic boundary conditions the modifications are:
H ( v i , v j ) = v j bc ( t − τ ) − v i ( t ) (4.34)
with the components of j bc obe ying the grid periodicity ,
j x , bc = ( ( j x − 1 ) mod n x ) + 1 , (4.35)
j y , bc = ( ( j y − 1 ) mod n y ) + 1 . (4.36)
Additi ve shifts are included in the c yclic modulo function to account for the ro w and column
indices starting at a node inde x of
1
instead of
0
. Preliminary simulations with either
boundary conditions,
( 4.33 )
or
( 4.34 )
, sho w the same patterns in the bulk area. This excludes
the possibility of spiral wa ve chimeras being a boundary induced artif act.
Note that instead of a phase frustration parameter
α
, as in the Kuramoto phase oscillator
model
213
, we employ time delay
245 , 295 , 329
. It can be shown, that small time delay
τ
plays a
similar role as
α
in coupled phase oscillators
221
. While
α
of fsets the phase difference in the
sinusoidal interaction function,
sin ( ϕ j ( t ) − ϕ i ( t ) − α ) , (4.37)

4.3 Numerical Simulations 61
Figur e 4.5 | Initial condition f or spiral wa ve chimeras.
(a) The initial phase distrib ution contains
a topological defect in the form of a phase singularity
87
at its center . At this point, all phases
coincide. (b) From the periodic beha vior in the time series of concentrations
v
(yello w) it is possible
to (c) calculate a phase v ariable ϕ , that increases linearly from 0 to 2 π between consecuti ve peaks.
the time delayed phase
ϕ j ( t − τ )
can be linearized around v anishing delay
τ ≈ 0
, such that is
also leads to a phase of fset:
sin ( ϕ j ( t − τ ) − ϕ i ( t ) ) , (4.38)
= sin ( ϕ j ( t ) − ϕ i ( t ) − τ d ϕ j ( t )
d t ) . (4.39)
Comparing
( 4.37 )
and
( 4.39 )
re veals
α = τ ω j ( t )
. Thus, in the time-delayed case the offset
depends on the rotation frequency
ω j ( t )
. In both cases the of fset makes it more dif ficult to
attain stable in-phase synchronization, because the interaction does not v anish in this state.
Ev en though it has been demonstrated that it is also possible to encode phase frustration
in the coupling coef ficient matrix
330
, using time delay
τ
is a more intuiti ve option, as it
naturally arises due to finite propagation speeds. Besides changing the chemical reaction
kinetics, the coupling coef ficients
C v → v = 2
and
C v → u = 1 / ε
, which are the prefactors of the
light intensity I i in ( 4.30 ), are not independently accessible.
Chimera states are kno wn to depend very sensiti v ely on initial conditions
272
. Ho we ver , it
turns out that the spiral wa ve v ariant can be initiated very reliably . Inspired by the traditional
way of initializing spiral w a ves, an earlier approach
245
required a meticulous protocol
that started with breaking a planar wa ve in a reaction-dif fusion system. Subsequently the
interaction type was morphed incrementally from local dif fusion to nonlocal coupling by

62 Chapter 4
slo wly increasing the maximum coupling range
l
and time delay
τ
. Since this procedure
depends crucially on the occurrence of a planar w av e and furthermore on the right duration
of each incremental stage, its success rate is not v ery high. Another approach
282
requires no
specialized e xternal interference: Spiral wa ve chimer as de velop by chance from a random
initial condition. T esting this procedure in simulations with the ZBKE model
( 4.30 )
re vealed,
that the probability for a successful spiral initiation was not high enough, in order to be applied
in the chemical e xperiment. The final and most reliable procedure applies a non-v anishing
topological char ge
Q  = 0
that is a feature of spiral wa ves
38
. Instead of concentrations
u i
and v i , the initial condition is encoded in phases 311 ,
ϕ ic ( i x , i y ) = arctan ( i y − i y , 0
i x − i x , 0 ) , (4.40)
and contains a phase singularity at its center
( i x , 0 , i y , 0 )
(Fig. 4.5 a). The function
arctan
is the
four -quadrant in verse of the tangent function
tan
. It produces a smooth gradient from
0
to
2 π
while tra v ersing a unit circle along the azimuthal direction. For its successful application
in the numerical simulation, the mapping
ϕ → ( u , v )
is required, such that
( 4.40 )
can be
utilized for limit c ycle oscillators. Because there is no analytic parametrization of the limit
c ycle in the ZBKE model
( 4.30 )
, the alternati v e is to measure the concentrations during a
full oscillation c ycle and assign a phase v ariable to them, that increases linearly with time
(Fig. 4.5 b,c). In the chemical e xperiment the initial phases can be set by utilizing indi vidual
periodic forcing as described before
( 4.29 )
. Even with modest amounts of heterogeneity
or noise, a spiral wa v e chimera alw ays de v elops from initial condition
( 4.40 )
for suitable
parameter combinations.
The simulations re veal the e xistence of spiral wa ve chimeras o v er a lar ge range of coupling
parameters
K
and
τ
(Fig. 4.6 ). A spiral wa ve chimera is reliably detectable by e v aluating the
global topological char ge
Q = ∑ i Q i
, where the local charge
Q i
is calculated via a discretized
line inte gral 331 :
Q i = ∑
k
mod ( ∆ φ k , 2 π ) . (4.41)
The inde xed phase dif ferences
∆ φ k
are e valuated along a discretized loop enclosing oscil-
lator
i
. Measured far from the core re gion, a spiral wa ve as well as a spiral w av e chimera
e xhibits a topological charge of Q = ± 1 depending on the direction of rotation.
The detection of a spiral wa ve chimera is indicated in figure 4.6 with colored squares. If the
standard de viation of the topological charge fluctuates
( σ Q > 0 . 01 )
, no stable spiral wa ve

4.3 Numerical Simulations 63
Figur e 4.6 | Numerical phase diagram in the K- τ plane.
Spiral wa ve chimeras are detected if
the topological char ge
( 4.41 )
follo ws
| Q | = 1
and its standard de viation is
σ Q < 0 . 01
. Parameter
combinations which lead to spiral wa ve chimeras are colored from red to blue depending on the time-
a veraged dif ference between the mean periods of oscillators inside,
T core
, and outside the core,
T spiral
.
Gray squares indicate the absence of a spiral wa ve chimera. Fixed parameter v alues as in table 4.1 .
pattern e xists and the square is colored gray . Otherwise, the square is colored according to the
dif ference of the av eraged periods inside the core,
T core
, and outside of it,
T spiral
. This serv es
to highlight the period dif ference of oscillators in the coherent and incoherent parts, which
is characteristic for chimera states
332
. It turns out that in contrast to phase oscillators
311
,
spiral wa ve chimeras in the ZBKE model
( 4.30 )
feature a core region with slo wer as well
as faster oscillators than its surroundings. In agreement with pre vious findings for phase
oscillators
311
the o verall e xistence re gion for spiral wa ve chimeras e xtends from zero to
small v alues
τ < 0 . 25 T 0
of time delay
τ
. Furthermore, the interv al of delay v alues in which
chimeras are observ able shrinks for larger coupling strength K .
As a consequence, further exploration of parameter space is focused on a small v alue of
coupling strength
K
. K eeping
K
fix ed at
K = 5 . 25 × 10 − 4
, the impact of time delay
τ
is
e xplored for v alues spanning the natural rotation period from
τ = 0
to
τ = T 0
(Fig. 4.7 ).
As the time delay
τ
is increased, a sequence of spiral wa ve chimeras, antiphase clusters,
global in-phase synchronization and e ventually spiral wa ve chimeras ag ain is observed on
the nonlocally coupled array of ZBKE oscillators
( 4.30 )
. Representati ve samples from this
numerical phase diagram will later be plotted and compared to their e xperimental counterparts
sho wn in the next section (Figs. 4.10 and 4.11 ).

64 Chapter 4
Figur e 4.7 | Numerical phase diagram f or time delay τ .
The sequence of patterns on the delay-
coupled nonlocal grid comprises: Spiral wa ve chimera with slo w core (bright green), ordinary spiral
wa ves with no core (medium green), spiral wa ve chimera with f ast core (dark green), core splitting and
fleeting coherence as discussed in the next section (orange), antiphase clusters (red), coe xistence of
in-phase and antiphase clusters (orange), in-phase synchronization (yello w), spiral wa v e chimera (dark
green), core synchronization, as described in the next section (orange) and in-phase synchronization
again (yello w). The periods outside
T spiral
and inside the core
T core
are gi v en as solid black lines and
solid gray lines, respecti vely . In cluster regions only one of both periods is sho wn, because the other is
irrele vant. The dashed gray line represents the number of oscillators far from zero-lag synchronization.
T ime delay
τ
and periods
T spiral
,
T core
are normalized by the reference period
T 0
and the size
A core
is
normalized by the total number of oscillators in the array
N
. T ime delay range is resolved in 350
steps. Fixed parameter v alues as in table 4.1 .
At each v alue of time delay
τ
the stability of the spiral and cluster patterns is e v aluated by
starting from three suitable initial conditions. For spiral w a ve chimeras the initial condition
is gi ven by a phase distrib ution with a phase singularity
( 4.40 )
, whereas the
d
-clusters with
d = 1 , 2
are started from
d
-clusters o verlayed with a small amount of noise. The case
d = 1
corresponds to in-phase and
d = 2
to antiphase synchronization. After 60 periods
the perse verance of an initial state is e v aluated by checking if the topological char ge fulfills
Q = ± 1 for spiral wa ve chimeras or whether the generalized K uramoto order parameter 333 ,
R d = 1
N ⏐ ⏐ ⏐ ⏐ ⏐
N
∑
j = 1
e i d ϕ j ⏐ ⏐ ⏐ ⏐ ⏐
, (4.42)
satisfies
R d > 0 . 7
for
d
-clusters. The total number of oscillators in the grid is gi ven by
N = n x × n y
. This analysis re veals (Fig. 4.7 ), that the global in-phase synchronized state is

4.3 Numerical Simulations 65
stable for the entire range of time delay v alues (yello w , green, orange shading), e xcept for a
small windo w
τ ∈ [ 0 . 21 , 0 . 35 ] T 0
. In this interval only antiphase-synchronized states pre v ail
(red shading). Finally , in green regions the in-phase synchronized state coe xists with the
dif ferent variants of the spiral w a v e chimera, as is typical for chimera states 272 .
Further details of the chimera states are re v ealed by e v aluating the mean period inside
T core
and outside the core
T spiral
as well as its size
A core
. Depending on a discretized local Kuramoto
order parameter ,
R i = 1
| Ω R , i | ⏐ ⏐ ⏐ ⏐ ⏐ ⏐
∑
j ∈ Ω R , i
e i ϕ j ⏐ ⏐ ⏐ ⏐ ⏐ ⏐
, (4.43)
oscillators are declared as elements of the core
R i < 0 . 4
or of the spiral
R i > 0 . 6
. The set of
oscillators Ω R , i are defined similar to the square domain in equation 4.28 ,
Ω R , i = { i , j ∈ Z 2 ⏐ ⏐ ⏐ ∥ j − i ∥ max ≤ ℓ R } , (4.44)
with
ℓ R = 2
. Comparing both periods
T core
and
T spiral
sho ws (Fig. 4.6 ), that spiral wa ve
chimeras e xisting for small time delay
τ
can be di vided into three qualitativ ely distinct
subclasses, where the core oscillators ha ve a slo wer
T core > T spiral
, approximately equal
T core ≈ T spiral
or faster period
T core < T spiral
. Similar to pre vious results for phase oscilla-
tors
311
, the core size of fast core spirals gro ws with increasing time delay for
τ ∈ [ 0 . 05 , 0 . 18 ] T 0
.
The gro wth continues throughout the orange transition domain until the core fills the entire
grid (red domain). Here,
τ ∈ [ 0 . 21 , 0 . 35 ] T 0
, the oscillators are not incoherent, b ut instead
co ver the grid with a random arrangement of oscillators in antiphase. F or larger time delay ,
τ ∈ [ 0 . 39 , 0 . 72 ] T 0
, antiphase clusters are superseded by global in-phase synchronization,
where all oscillators share the same phase (yello w domain). Remarkably , for large time delay ,
τ ∈ [ 0 . 72 , 0 . 935 ] T 0
, there is a resur gence of spiral wa ve chimeras with a f ast core (green
domain). Note that the dynamical behavior does not repeat itself at a delay v alue of the full
natural period
τ = T 0
, because spiral wa ve chimeras transition into in-phase synchronization
(yello w domain) around τ ≈ 0 . 94.
The preceeding numerical e vidence (Figs. 4.6 and 4.7 ) strongly suggests the possibility of
finding spiral wa ve chimera states in the chemical e xperiment with initial condition
( 4.40 )
for suitable coupling parameters. Furthermore it will be sho wn in the next section ho w the
numerical findings on cluster states can be rationalized with the peculiar shape that the phase
response curv e assumes for strong coupling.

66 Chapter 4
4.4 Results
Figur e 4.8 | Experimental observation of a spiral wa ve chimera state.
(a) Spiral wa ve chimera
in an array of
N = 1600
photochemically coupled BZ oscillators. The gray value pattern
v i
is
from fluorescent light emitted by the reduced catalyst
Ru(dmbpy) 2+
3
. The spiral rotates with a
period
T spiral = 33 s
around the incoherent core consisting of approximately
40
phase-randomized
oscillators. Image taken at
t = 700 s
after initiation. (b) Oscillator phases obtained from the gray
v alues measured in (a). (c) Periods of the oscillators in (a) illustrating that oscillators in the spiral wa ve
are approximately frequency synchronized in the rotating spiral w a ve, while the asynchronous core
oscillators exhibit shorter periods. (d) Space-time plot of the spiral wa ve chimera from measurements
along a cross section
x core ( t )
, that follo ws the core center during 14 rotational periods of the spiral.
The coloration combines information from grayv alues
v i
(a), which select the brightness, and the local
order parameter
R i ( 4.43 )
, which determines the color . Coupling parameters in
( 4.25 )
are:
K = 0 . 08
,
κ = 3 . 1
,
τ = 2 s
,
I 0 = 0 . 06 mW cm − 2
,
ℓ = 4
. Initial reactant concentrations:
[H 2 SO 4 ] 0 = 0 . 77 M
,
[NaBrO 3 ] 0 = 0 . 51 M , [NaBr] 0 = 0 . 08 M , [malonic acid] 0 = 0 . 16 M .
W ith the experimental setup described in section 4.2 and an informed choice of coupling
parameters (section 4.3 ), we successfully confirmed the e xistence of spiral wa ve chimeras
e xperimentally
334
(Fig. 4.8 ). The chemical oscillators for the e xperiment are selected from a
reserv oir of
2816
beads, whose period distrib ution has a mean of
( 78 . 3 ± 23 . 6 ) s
. Selecting
suitable beads based on their periods from the interv al
T ∈ [ 70 , 98 ] s
, results in a narro w
period distrib ution with a mean of
⟨ T ⟩ = ( 85 . 7 ± 7 . 1 ) s
and a width of
δ T = 8 . 3 %
relati ve
to the mean period. T aken together the oscillators constitute a two-dimensional grid netw ork
consisting of N = 40 × 40 = 1600 nodes.

4.4 Results 67
The initial condition
( 4.40 )
is enforced with indi vidual periodic forcing
( 4.29 )
for a duration
of
200 s
. On the two-dimensional grid this results in the feedback taking the shape of a
rotating, triangular -shaped wa ve with one corner centered at
( x 0 , y 0 ) = ( 20 , 20 )
. T o guarantee
entrainment, the forcing signal has a rectangular wa veform and a period
T forcing = 48 s
that is
smaller than the lo wer bound of the selected period interval.
Directly afterwards, nonlocal photocoupling
( 4.25 )
between oscillators is initiated. During
the first two periods the spiral w a ve formed while the core turned incoherent. Over the
e xperiment, the size of the core fluctuated between
20
and
40
oscillators. A snapshot at
t = 700 s
of the fluorescence intensity emitted by the chemical beads and measured as
grayv alues
v i
by the camera is sho wn in figure 4.8 a. T o quantify synchronization with the
local K uramoto order parameter
( 4.43 )
, oscillator phases
ϕ i
(Fig. 4.8 b) are calculated by
linear interpolation between two consecuti v e peaks in the grayv alue time series (Fig. 4.5 b,c)
via equation
( 3.10 )
. Grayvalues and phases clearly e xhibit the distinctiv e incoherence in
the core and the coherence of the spiral w av e surrounding it. Both re gions exhibit dif ferent
mean periods,
T spiral  = T core
, (Fig. 4.8 c), which is another characteristic feature of chimera
states 332 .
While the snapshots depict the spatiotemporal pattern at a fix ed instant in time, the temporal
e volution is visualized with a k ymograph or space-time plot (Fig. 4.8 d). Here, we employ an
enhanced k ymograph, that simultaneously of fers information on the spatial structure of the
pattern as well as the local le v el of order
( 4.43 )
. The grayv alues
v i
, which encode the spatial
details of the pattern, contrib ute to the brightness and the local K uramoto order parameter
R i
determines the color . This has the adv antage, that the incoherent firing e v ents in the core
and the coherent rotating wa ve are resolv ed and colorized simultaneously . In addition, the
k ymograph is not based on a fixed cross section
x core
, b ut tracks the current core center . This
ensures that the object of interest, in this case the core, is not outside the cross section. This
re veals that the spiral wa ve chimera shares the characteristic feature of spiral wa ves: The
core re gion alternatingly emits wa v es in opposite directions. Eventually the motion of the
spiral wa ve chimera leads to its termination. In the experiment the core collided with the
boundary of the grid after lasting for 38 rotation periods.
The tip of a spiral wa ve in a reaction-dif fusion system can be tracked by computing the
intersection point of dif ferent iso-concentration lines
337
or the cross product of concentration
gradients
338
. Due to the non-smooth, incoherent core, these approaches are not applicable to
spiral wa ve chimeras. Instead the local order parameter
R i
is utilized for reliable tracking. All
oscillators
i
with
R i ≤ R threshold
are collected for
R threshold = 0 . 4
. Then each node is assigned
a weight
w i = 1 − R i / R threshold
that increases with declining order . Finally , the potentially
multiple core positions are calculated as the weighted centroids
r core = ∑ i w i i
of each simply-

68 Chapter 4
Figur e 4.9 | Core trajectory .
(a) Unlike a re gular spiral wa ve, whose tip performs rigid rotation or
meandering
335
, the center of the incoherent core (Fig. 4.8 ) performs erratic motion as highlighted
by the superimposed white line. The starting and end points are marked with a black-dotted and an
unfilled circle, respecti vely . (b) In a logarithmic plot the mean-square displacement sho ws a roughly
linear increase as a function of the measurement time interv al
∆ t
. The corresponding scaling exponent
was determined from a linear fit (red) as
1 . 05
, which is characteristic for erratic Bro wnian motion
336
.
connected set of lo w-order oscillators. This method has prov en robust ag ainst core size
fluctuations and e ven works reliably in cases when a single core splits into man y cores. The
tracking algorithm is successfully employed (Fig. 4.9 ) to follo w the core position of the
e xperiment presented in figure 4.8 . The starting point of the trajectory (white line in figure
4.9 a) at
( x 0 , y 0 ) = ( 20 , 20 )
is marked with a white circle o verlaid with a black dot. Instead of
rigid rotation or meandering
339
the core of the chimera spiral wa ve tra verses an irre gular path.
This random motion is quantified with the mean square displacement
336
(MSD), which is a
measure for the erratic motion of indi vidual particles. It is calculated from a single trajectory
by a v eraging in time ov er all squared displacements,
∆ r 2 ( ∆ t ) = ∆ x 2 ( ∆ t ) + ∆ y 2 ( ∆ t ) , (4.45)
occurring for increasing time interv als ∆ t :
MSD ( ∆ t ) = ⟨ ∆ r 2 ( ∆ t ) ⟩ t = 1
N
N
∑
i = 1
∆ r 2
i ( ∆ t ) . (4.46)
Applied to the e xperimentally recorded core trajectory , this re veals a linearly increasing
MSD (Fig. 4.9 b). The corresponding scaling exponent is
1 . 05
, which is an indicator of
dif fusiv e Bro wnian motion
336
. The random walk of the core is in agreement with pre vious
observ ations in one 340 , 341 and two dimensions 245 , 311 .

4.4 Results 69
In the case of spiral wa ve chimeras, the random mo vement w as suspected
311
to be a conse-
quence of the neutral stability of spiral wa ves with re gards to translational perturbations 339 .
It is kno wn that random forcing leads to erratic motion of spiral wa v es
342
. Furthermore a
spiral wa ve is most vulnerable in the tip re gion
133
, which is where the incoherent core of
a spiral wa ve chimera is located at. T aken together , the random perturbations due to the
incoherent core are likely to dri ve the random motion of the spiral w a v e chimera.
Be yond the experimental v erification of spiral wa ve chimeras, we further in vestig ate what
patterns can emer ge on an array of nonlocally delay-coupled oscillators (Figs. 4.10 and 4.11 ).
Guided by the e xploratory numerical simulations (Fig. 4.7 ), we successfully found represen-
tati ve e xamples for e v ery distinct dynamical beha vior in the chemical experiment.
At small v alues of time delay
τ
, three spiral patterns e xist, that can be dif ferentiated by
the beha vior of the core oscillators. In the order of increasing delay
τ
, first spiral wa ve
chimeras appear whose oscillators are slo wer in the core than the spiral wa ve
T core > T spiral
(Fig. 4.10 a,b). The preliminary numerical simulations (Fig. 4.6 ) suggest, that this behavior is
more pronounced for lar ge coupling strength
K
, so in the e xperiment we choose a small time
delay of
τ = 1 s
with a lar ger
K = 0 . 15
than pre viously (Fig. 4.8 ). The period heterogeneity is
T 0 = ( 54 . 8 ± 1 . 4 ) s
. The slow core only consists of a small, but v arying number of oscillators,
A core = 2 − 10
with
T core = 49 . 2 s > T spiral = 31 . 3 s
. The slo w-do wn is due to the constant
perturbations from the surrounding rotating wa ve, which ef fecti vely increase the illumination
le vels
I i
, such that the periods
T i
are prolonged. It drifts slo wly on the grid until it e ventually
v anishes after collision with a boundary after 61 rotation periods.
In a small interv al of time delay
τ
spiral wa ves e xist, whose core oscillators are neither
incoherent nor are their periods dif ferent from the surrounding spiral wa ve (Fig. 4.7 and
Fig. 4.10 e,f). The y exist at the intersection point, where decreasing core periods
T core ( τ )
and
rising spiral wa ve periods
T spiral ( τ )
match as the time delay
τ
is increased. In the e xperiment
the y were found for K = 0 . 1, κ = 2 . 5, T 0 = ( 114 . 8 ± 7 . 2 ) s and τ = 2 s.
The spiral wa ve chimera with a f ast core
T core < T spiral
(Fig. 4.8 and Fig. 4.10 e,f) is the
most pre valent representati ve in the experiments as its e xistence domain is the largest for
small coupling strength
K
(Fig. 4.7 ). Besides the example pre viously discussed (Figs. 4.8
and 4.9 ), it was also observ ed for v alues of delay
τ = 3 s
and
τ = 5 s
. Note that in all cases,
the spiral wa ve chimera terminates by collision with a boundary . After its disappearance,
global in-phase synchronization emer ges. This verifies the characteristic coe xistence
332
of
the chimera state and in-phase synchronization.

70 Chapter 4
Figur e 4.10 | Overview of patter ns in nonlocally coupled oscillator arrays - Part I.
For increasing v alues of time delay
τ
we find the follo wing
sequence of patterns in chemical experiments (top ro w) and in numerical simulations (bottom ro w): (a,b) Spiral wa ve chimeras with a slo w core,
(c,d) spiral wa ves with no core , (e,f) spiral wa ve chimeras with a fast core and (g,h) core splitting. The series continues (Fig. 4.11 ) on the next page.

4.4 Results 71
Figur e 4.11 | Over view of patterns in nonlocally coupled oscillator arrays - P art II.
Continuing from the pre vious page (Fig. 4.10 ), we find in
chemical experiments (top ro w) and in numerical simulations (bottom ro w) for increasing v alues of time delay
τ
: (i,j) Fleeting coherence, (k,l)
antiphase clusters, (m,n) an isochronous cluster and (o,p) spiral wa ve chimeras, whose f ast core oscillators synchronize in-phase at the transition to
in-phase synchronization. This sequence sho ws representati ve simulations analyzed in figure 4.7 .

72 Chapter 4
Figur e 4.12 | Core Splitting .
(a)
t = 4 T 0
: An initially small incoherent core of a spiral wa v e chimera
gro ws, until (b) at
t = 19 T 0
the core has more than tripled in circumference. (c)
t = 26 T 0
: The spiral
tip is so slo w in completing a lap around the core, that it enables the high-frequency core to emit a
ne w wa ve se gment. This se gment introduces two ne w spiral tips, which are highlighted with a thick
yello w annulus in contrast to a thin annulus for the pre-existing tip. (d)
t = 31 T 0
: Afterwards a part of
the incoherent core synchronizes in frequency and phase to the surrounding oscil lators, which di vides
the core into two parts. (e)
t = 31 T 0
: At the end of the splitting e vent there are three tips rotating
around two cores, which conserv es the total topological char ge in the system
38
. (f)
t = 140 T 0
: In a
lar ger grid with periodic boundary conditions
( 4.34 )
, splitting e v ents continue until merging e vents
balance the number of cores. The final state resembles a v ortex glass 343 .
In the transition re gion between spiral wa v e chimeras and antiphase clusters, the core
size
A core
e xhibits a pronounced increase (Fig. 4.7 ). Due to the increased core size, it takes
longer for the spiral tip to complete a c ycle around the core (Fig. 4.12 a.b). This allows
for the nucleation of a wa ve se gment at the far side of the core relati ve to the current tip
position (Fig. 4.12 c). Note that the total topological charge
Q ( 4.41 )
is conserv ed during the
nucleation, because the open ends of the wa ve se gment are equi valent to tw o new spiral tips
with opposing chiralities,
Q after = + 1 − 1 + 1 = + 1 = Q before
. Shortly afterwards, a portion
of the core oscillators frequenc y-locks to the surrounding spiral wa ve (Fig. 4.12 d), such that
the core fractures into two disjoint entities. In total the number of cores increased by one and
the number of tips by two during the splitting e vent.
In later stages, further ne w cores nucleate via splitting while old existing ones can mer ge
together or collide with boundaries. The number of cores e ventually reaches a stationary size,

4.4 Results 73
when the array is cov ered with the largest number of cores it can support simultaneously .
The spatial core arrangement (Fig. 4.12 f) does not exhibit long-range crystal-lik e order , b ut
still retains the characteristic short-range order of an amorphous structure . A similar pattern
observ ed with spiral wa ves in a reaction-dif fusion system was termed v ortex glass 343 .
The observ ed beha vior (Fig. 4.10 g,h) is unlike similar spiral w a ve instabilities such as spiral
breakup
344 , 345
, where the wa v e speed depends on curv ature in such a way that backfiring
in the core region leads to turb ulence. Also Kapral et al. observed a similar scenario
106
for
spiral wa ves on a coupled map lattice. In their case chaotic dynamics, that are localized at
the wa ve front, induce transv ersal wa ve instabilities 346 .
F or slightly larger v alues of delay the core expands too quickly to allo w for splitting, but
does not co ver the entire array . The resulting pattern is a mixture of fleeting, intertwined
coherent and incoherent domains (Fig. 4.11 i,j).
At lar ge enough time delay
τ
(Fig. 4.7 ), spiral patterns are superseded by antiphase
d = 2
(Fig. 4.11 k,l) and in-phase
d = 1
clusters (Fig. 4.11 m,n). The phase dif ference between
oscillators constituting dif ferent clusters is
∆ ϕ d = 2 π / d
. Their spatial arrangement depends
sensiti vely on initial conditions. For an initial phase distrib ution in the shape of a target w av e,
the clusters arrange in approximate concentric circles (Fig. 4.11 k,l), while for random initial
conditions the y are placed irregularly . The periods
T d
of the clusters are linear functions of
the time delay ,
T d ( τ ) ∼ d · τ
, whose slopes are gi ven by the number of clusters
d
. Note that
in-phase synchronization is v ery rob ust, since it e ven emer ged in an experiment with v ery
heterogeneous oscillators, whose mean period was
T = ( 75 . 3 ± 21 . 5 ) s
. This amounts to a
spread of the period distrib ution that was 28 . 6 % relati ve to the mean period.
F or large time delay
τ
, spiral wa v e chimeras emer ge again (Fig. 4.7 ). Their core oscillators ex-
hibit a faster oscillation c ycle than the surrounding spiral wa ve
T core < T spiral
. Ho we ver , the y
do not transition to antiphase clusters, b ut instead to in-phase synchronization. This transition
in volv es the formation of in-phase synchronized patches inside of the core (Fig. 4.11 o,p).
These patches gro w and push the wa v e rotating around them into the boundary where it
annihilates. When the value of the time delay coincides with the unperturbed oscillation
period,
τ = T 0
, in-phase synchronization dominates with a period that is half of the time
delay , T ( τ ) ∼ 0 . 5 τ .
The mechanism behind cluster formation and their periods can be understood from the phase-
resetting character
87 , 347 – 349
of the particular BZ oscillators, b ut extends to general strongly
coupled relaxation oscillators. In the absence of external perturbations, the phase
ϕ
of an
oscillator can be defined in such a way that it increases uniformly
( A.4 )
during an oscillation
period (see appendix A.1 ). In response to a perturbation, the phase
ϕ
may be repulsed,

74 Chapter 4
Figur e 4.13 | Comparison of experimental and numerical phase response cur ves.
(a) Simulta-
neous measurements (full circles) of the phase change
∆ ϕ
on a reserv oir of oscillators at v arying
phases for dif ferent amplitudes
I p
of the externally applied light perturbation:
0 . 09 mW cm − 2
(yello w),
0 . 25 mW cm − 2
(red) and
0 . 62 mW cm − 2
(purple). The background intensity is
I 0 = 0 . 06 mW cm − 2
.
Full lines are piece wise fits
( 4.48 )
to the experimental data. Shaded areas are bounds for all datapoints
corresponding to a gi ven perturbation amplitude. (b) Perturbations at early phases in the limit cycle
lead to a v anishingly small phase recession
∆ ϕ ≲ 0
. Perturbations at a later stage induce a phase reset,
such that the oscillation cycle restarts. (c) Numerically obtained PRCs for ZBKE oscillators
( 4.30 )
are in e xcellent agreement with the experimentally measured counterparts in (a). The results for three
dif f erent perturbation amplitudes
I p
are plotted:
0 . 2 × 10 − 3
(yello w),
2 . 5 × 10 − 3
(red) and
5 . 0 × 10 − 3
(purple). The background illumination intensity is
I 0 = 5 . 25 × 10 − 4
. (d) The phase response curve
of a strongly perturbed FitzHugh-Nagumo oscillator
( C.7 )
resembles its counterpart of the ZBKE
oscillator in (c). The perturbation amplitudes I p are 0 . 4, 1 . 5 and 2 . 0.
∆ ϕ < 0
, or adv anced,
∆ ϕ > 0
. It is possible to quantify the ef fect of a gi ven perturbation by
directly measuring the resulting period change from the unperturbed period
T 0
,
∆ T = T 0 − T p
,
due to applied perturbations at v arious phases
ϕ
during the oscillation c ycle. The period
change ∆ T is translated into a phase change ∆ ϕ ,
∆ ϕ = 2 π
T 0
∆ T , (4.47)
which results in the phase response curve (PRC)
∆ ϕ ( ϕ )
. Measurements of the PRC for
a short rectangular intensity perturbation
I 0 → I 0 + I p
are sho wn in figure 4.13 . Results
from chemical e xperiments and simulations with the ZBKE model
( 4.30 )
are in e xcellent
agreement. They sho w that a perturbation at the beginning of the oscillation c ycle very
slightly delays the phase
∆ ϕ ≲ 0
. At a certain critical transition point
ϕ ∗ ( I p )
, which depends

4.4 Results 75
on the perturbation amplitude
I p
, a discontinuous jump occurs. Be yond the jump point,
perturbations reset the oscillation c ycle by inducing an immediate ne w spike
87 , 350
. Fitting
the e xperiments with a piece wise linear function,
PRC ( ϕ ) = ⎧
⎨
⎩
0 , ϕ < ϕ ∗ ( I p )
2 π + m · ϕ , ϕ ≥ ϕ ∗ ( I p ) (4.48)
re veals, that the PRC in the second part decays with a slope of
m = − 1
, since the current
phase
ϕ
changes by the remaining phase dif ference to
2 π
. For weak perturbations the fit is
biased by v arying jump points of heterogeneous chemical oscillators (Fig. 4.13 a).
This beha vior has also been observ ed in v arious biological settings, such as canine Purkinje
fibers
351
, galline and leporine cardiac sinoatrial pacemak er cells
352 , 353
, interneurons respon-
sible for the respiratory c ycle in lamprey fish Lampetra fluviatilis
354
, electrosensory pathway
neurons in electric fish Eigenmannia
355
, pyloric pacemaker neurons in lobsters Homarus
Americanus
356
under electrical perturbation, circadian rhythm in Homo sapiens
357
, algae
Gon yaulax polyedra 358 , fruit flies Drosophilia melanogaster 359 under light-stimulus.
Such discontinuous phase response curves were cate gorized as type zero by W infree
87
and are characteristic for strongly perturbed oscillators. In contrast to weakly perturbed
oscillators
213
, the phase response curv e does not scale linearly with the perturbation strength
K ,
PRC ( K , ϕ )  = K · PRC ( ϕ ) , (4.49)
b ut changes its shape. In the case of BZ oscillators (Fig. 4.13 ), the jump point
ϕ ∗
mo ves to
smaller phases for lar ger perturbation strengths.
The shape of the PRC and its dependence on the perturbation strength (Fig. 4.13 ) are direct
consequences of the underlying phase portrait (Fig. 4.14 a). While an oscillator tra v erses
the limit c ycle in the ZBKE model
( 4.30 )
, it mov es with high speed during the e xcitation
and e xtremely slo wly during recov ery , when it is on the left half of the attractor . A light
perturbation
I → I + ∆ I
is equi valent to an additi ve of fset on the current oscillator position.
This of fset is predominantly in the horizontal direction, because the light prefactor in the first
equation of the ZBKE model is lar ger than in the second one,
1 / 0 . 11 > 2
. If the perturbation
is small, the state returns after a brief e xcursion through a slo w phase space re gion. Lar ge
perturbations push the oscillator be yond the black
u
-nullcline
u 0
(Fig. 4.14 a), such that
the y can not directly return without a large e xcursion in phase space. This explains why
perturbations early in the limit c ycle are of little ef fect. During the excitation, states v ery

76 Chapter 4
Figur e 4.14 | Phase space dynamics underlying phase reset beha vior .
(a) The phase portrait of
the ZBKE model
( 4.30 )
exhibits a stable limit c ycle (yellow) around an unstable fix ed point (unfilled
circle) at the intersection of the gray , dashed
v
-nullcline
v 0 ( u )
and black, dashed
u
-nullcline
u 0 ( u )
.
Y ello w arrowheads indicate the f ast phase flo w on the right part of the limit c ycle and the v anishingly
slo w one on the left part. A perturbation in
u
-direction that induces a phase reset is indicated with a
red arro w . (b) The distance
∆ u LC − u 0
between the limit cycle and the black
u
-nullcline
u 0
at v arying
phases
ϕ
(orange, dashed) directly coincides with the phase jump point
ϕ ∗
in PRCs for positi ve
rectangular perturbations in
u
for dif f erent perturbation amplitudes (blue). The red dot corresponds to
the position and size of the red arro w in (a).
quickly return to the right part of the limit cycle. Once the state reaches the slo w , right half of
limit c ycle, perturbation-induced excursions are possible, b ut the distance from the limit cycle
to the nullcline
∆ u L C − u 0
is v ery large for early phases. At later phases the distance
∆ u L C − u 0
decreases e xponentially . Hence a smaller perturbation can more easily cause an excursion,
which is equi valent to resetting the phase and initiating a ne w spike. This mechanism is
reasonable, because the distance
∆ u L C − u 0 ( ϕ )
as a function of phase
ϕ
along the limit c ycle
is identical to the phase jump point
ϕ ∗
in PRCs for perturbations of dif ferent amplitudes in
u
-direction (Fig. 4.14 b). Note that on a logarithmic scale the ZBKE model
( 4.30 )
resembles
the FitzHugh Nagumo (FHN) model
( C.7 )
, which is the prototypical model for neuronal
e xcitability type II
348
. Since the phase response curv es of the ZBKE and FHN models agree
v ery well (Fig. 4.13 d), the y are e xpected to gi ve rise to similar patterns on a nonlocally
coupled array of oscillators. Indeed, all behaviors disco vered with the ZBKE model (Fig. 4.7 )
were successfully reproduced with the FHN model
334
. Furthermore, core splitting (Fig. 4.12 )
seems to be a beha vior that is e xclusi vely associated with type zero phase resetting, because
it could not be reproduced in simulations with phase or Stuart-Landau 213 oscillators.

4.4 Results 77
Figur e 4.15 | Antiphase cluster states.
(a) Experimental snapshot of an antiphase pattern with one
group of oscillators at phase
π / 2
(blue) and the other
∆ ϕ = π
further at
3 π / 2
(orange). Oscillators
at the grid boundaries beha ve dif ferently , because they ha ve less nonlocal neighbors. (b) Schematic
mechanism of antiphase patterns, where reciprocal perturbations arriv e after a trav el time
δ = ∆ t rise + τ
.
(c) The total period of the cluster state is the sum of tra v el times
δ
that is required to pass the jump
point t ∗ = ϕ ∗ T 0 / 2 π in the PRC.
Antiphase patterns (Fig. 4.15 a) and more generally
d
-cluster patterns are a direct consequence
of strong, nonlocal coupling on an oscillator array . A strongly coupled oscillator responds
to perturbations in one of two w ays: It is either unresponsiv e or it fires. If a perturbation
due to nonlocal coupling
( 4.25 )
is too small to cause a phase-reset, it is possible to either
increase the coupling strength
K
or wait for a time
∆ t
until the oscillator phase
ϕ
increased
past its phase-reset point
ϕ ∗ ( K )
in order to successfully trigger a spike. In the case of an
array of oscillators with random initial phases, one part of the population will fire while the
other stays quiescent. Ho we v er , gi ven that the coupling strength
K
is suf ficiently large, the
unresponsi ve population will fire after a time interv al of
∆ t = δ ·
· = ∆ t rise + τ
. This is the
time duration for the perturbation from the group of the population that fired first to arri ve.
The time interv al
δ
is composed of
∆ t rise ≈ 1 s
, which is the time required for a peak in the
grayv alue time series to rise and the time delay
τ
in the coupling
( 4.25 )
. Continuing in this
fashion, the first group will fire again a time
∆ t = δ
after the second group (Fig. 4.15 b).
Ultimately this mechanism determines the periods of the oscillators in the array to be
T antiphase ( τ ) = 2 ( ∆ t rise + τ ) = 2 δ . (4.50)
The e xistence of the antiphase clusters depends on the coupling strength
K
. It is required to
be lar ge enough, such that the accumulated tra vel times during one period e xceed the jump
point t ∗ = ϕ ∗ T 0 / 2 π after which a phase-reset is triggered: 2 δ ≥ t ∗ ( K ) (Fig. 4.15 c).
It is straightforward to generalize the mechanism behind antiphase clusters to
d
-clusters
with positi ve inte gers
d ∈ N +
. A
d
-cluster may emer ge, if the time delay
τ
and coupling

78 Chapter 4
Figur e 4.16 | d-Cluster states.
(a) Theoretical prediction of periods (black line) and e xistence
interv als (colored background) of
d
-clusters. At the beginning of each interv al, the period drops
sharply and then increases with a slope of
d / ∆ d
. The delay time
τ
is scaled, such that the jump
point agrees with the v alue measured in simulations,
t ∗ = 30 . 2
. (b) Numerical simulations verify
the sequence of clusters and the qualitati ve beha vior of periods. Simulations are performed on an
array of
64 × 64
oscillators with periodic boundary conditions. Each
d
-cluster state is identified with
a localized version of
( 4.42 )
. The coupling strength
K = 5 . 25 × 10 − 5
was chosen small enough to
allo w for emer gence of
d
-clusters with
d > 3
. The theoretical approach fails to predict the dynamics
in the gray regions, which feature chimera spots and stripes, consisting of dif ferent
d
-cluster states. In
addition the first re gion with wa v e synchronization up to
τ = 5
was missed. Both shortcomings are
probably due to the ne gligence of oscillator heterogeneity and the phase-delaying part of the phase
response curve (Fig. 4.13 ).
strength
K
gi ve rise to a sequence of
d
perturbations, that exceed the jump point
t ∗
, such
that the y obey:
d · δ ≥ t ∗ ( K )
. Of all possible clusters satisfying this condition, the
d
-cluster
with the shortest firing sequence,
min ( d · δ )
, is established. This selection principle has an
intuiti ve reason: Oscillators in
d
-clusters with a longer firing sequence get recruited to the
fastest d -cluster o ver time.
In addition, it is possible that not all
d
oscillator clusters fire consecuti vely , b ut instead
∆ d
clusters are skipped at each e xcitation. In the resulting
d
-cluster period
T d ( τ )
, this can be
accounted for by di viding ov er the number of omitted clusters:
T d ( τ ) = d
∆ d ( ∆ t rise + τ ) . (4.51)

4.4 Results 79
Figur e 4.17 | Segmentation of the pattern in ph ysical space and phase space.
(a) Numerical
simulation of a spiral wa ve chimera, where the phases in the spiral w a ve b unch together , such that the
profile of the wa ve breaks into
d = 5
clusters. (b) Also in the
u
-
v
phase space of concentrations fi v e
distinct groups of oscillators aggreg ate on the slo w left part of the limit c ycle. (c) The segmentation
into fi v e separate groups is clearly visible on a circular phase histogram of core (blue) and spiral wa v e
oscillators (red). The core oscillators are attracted to the same se gments, but can switch between them.
F ollo wing the same approach, the existence interv als for the clusters (Fig. 4.16 a) can be
computed. For ∆ d = 1, the y assume a simple expression:
t ∗
d − 1 < τ + ∆ t rise ≤ t ∗
d . (4.52)
As sho wn in the comparison with numerical simulations (Fig. 4.16 b), the sequence of
d
-cluster states can be correctly predicted as a function of time delay
τ
. This result is also
in agreement with simulations of nonlocally coupled BZ oscillators on a ring
360
. Note that
the abo ve approach assumes global coupling between all oscillators. Howe v er , the two-
dimensional nonlocal coupling kernel with a side length of
2 ℓ + 1 = 11
takes into account
121 oscillators, which approximates global coupling.
Clustering also plays a role for the spiral pattern. The smooth wa v e profile di vides into
d
consecuti ve se gments. This occurs due to the same reason driving t he
d
-cluster states. In
e xperiments the segmentation is much less pronounced, since the jump points
ϕ ∗
are more
heterogeneous, than in the simulation (Fig. 4.13 ). Furthermore, the cluster mechanism also
e xplains the core synchronization at lar ge v alues of the delay (Fig. 4.11 o,p), where the
oscillators follo w a period of period
T = 0 . 5 ( τ + ∆ t rise )
. The delay history of the coupling
( 4.31 )
, which encompasses nearly a complete period
T 0
, contains peaks from the oscillator
itself and others at approximately half a period later . Over time all core oscillators share
the same delay history . As the core gro ws, e v entually all oscillators in the entire array are
entrained and in-phase synchronized.

80 Chapter 4
4.5 Short summary
W e establish that photocoupled catalyst-loaded microparticles in Belousov-Zhabotinsk y
solution form a v ersatile experimental setup for e xperiments with relaxation oscillators on
lar ge networks e xceeding N = 1000 nodes.
As a first e xample of the experimental capabilities we e xperimentally in vestigated and v erified
the spiral wa ve chimera
334
(Fig. 4.8 ), which was predicted by K uramoto in 2002
271
. Besides
v erifying its existence, we also de vised a core-tracking algorithm to characterize the erratic
motion of the core (Fig. 4.9 ).
T aking the time delay
τ
as a bifurcation parameter , we explored the phase diagram of a
two-dimensional array of nonlocally coupled oscillators numerically and e xperimentally
(Figs. 4.7 , 4.10 and 4.11 ). W e disco vered pre viously unreported patterns and transitions, such
as spiral wa ve chimeras with a slo w core (Fig. 4.10 a,b), core splitting (Figs. 4.10 g,h and
4.12 ) and core synchronization (Fig. 4.10 o,p). Apart from spiral patterns, we also observed
d
-cluster states and described a mechanism that e xplains their emergence based on time delay
τ
and the type zero phase response curv e (Fig. 4.15 ), that plays a role in neurobiology for
strongly perturbed nerv e cells 87 .
Our findings are of significance be yond the chemical BZ oscillator
( 4.30 )
, since they could be
reproduced in the canonical model for nerv e excitation, the FitzHugh Nagumo model
( C.7 )
.
Thus we e xpect our findings might be of rele v ance to cardiac and cortical dynamics.
The e xperimental setup opens the door for a lar ge number of future experiments. Some of
which are outlined in the ne xt section.

4.6 Future Directions 81
4.6 Futur e Dir ections
Figur e 4.18 | Spiral W a ve Chimera Contr ol.
(a) Spiral wa ve chimera after two laps on a forced
circular trajectory (white). (b) T ime-dependent nonlocal kernel with spatial asymmetry for orienta-
tion ( 4.53 ).
The e xperimental setup allo ws for a plethora of future e xperiments. W ithout further modifi-
cations of the e xperimental soft- or hardware, it would be possible to also do e xperiments
on further chimera states in one, two and three dimensions, such as spots and scroll w a ves.
Three dimensional grids would be limited to small systems, such as
10 × 10 × 10
oscillators.
Chimera scroll rings could be studied nonetheless in a reduced tw o-dimensional array by
e xploiting the azimuthal symmetry of the ring with reduced cylindrical coordinates.
On a one-dimensional ring dif ferent theoretically proposed control algorithms
302 – 305
can be
implemented and tested. Furthermore control schemes for spiral wa ve chimeras could be
de vised by generalizing their counterparts in one dimension. F or e xample it is possible to
e xtend the asymmetric kernel in one dimension used pre viously 303 to two dimensions
G ( r , φ ) = e − α ( φ − φ 0 ) r , (4.53)
α ( φ − φ 0 ) = α 1 + 1
2 ( α 2 − α 1 )( 1 + cos ( φ − φ 0 )) . (4.54)
Here the asymmetric kernel is written in polar coordinates with an orientation angle
φ 0
, which
can be e xternally manipulated to set the drift direction. The asymmetry of the kernel,
α 2 − α 1
,
determines the speed of the drift. In figure 4.18 the orientation angle increases linearly in
time, leading to two circular laps. W ithout control, the core location would perform a very
slo w Brownian motion around the array center .

82 Chapter 4
Another possibility is testing those schemes that ha v e prov en successful in the control of
ordinary spiral wa ves
125 , 166 , 167
. Further studies of spiral wa ve chimeras might re veal ho w
and if the y pin to spatial or parameter heterogeneities, like their re gular counterparts
361 , 362
.
Apart from chimera states, it would be interesting to v erify that domains of lar ge oscillator
heterogeneity act as natural pacemakers 363 .
One obstacle in the implementation of control algorithms will be the calculation of the
instantaneous phase v ariable. In the current system, the phase is only known retrospecti v ely ,
after the oscillator peaked ag ain. One way around it, is to use a dif ferent order parameter
306
,
that is not deri ved from the instantaneous phase ϕ ( t ) .
Be yond regular grids, chimera states can also be in v estigated in comple x networks, such as
small world
364
and scale-free
286
networks. In addition connectomes known from biology ,
such as the hermaphrodite soil worm Caenorhabditis ele gans
321
with 302 neurons, portions
of fruit fly Drosophilia melanogaster connectome
365
with 380 neurons or coarse-grained
macaque cortical network
366
can be implemented. In this venue it will be interesting
to see ho w neuronal time-dependent connectivity as shaped by spik e timing dependent
plasticity
174
contrib utes to the natural rise of chimera states as it reinforces synchronized
and desynchronized domains via potentiation and depression, respecti vely .
It seems feasible to experimentally v erify the explosi ve synchronization transition
367
as
well as symmetry cluster synchronization
202
by selecting nodes with an appropriate period
distrib ution. Re garding the requirements of finding them, it is already possible to run e xperi-
ments on arbitrary networks with selectable coupling strength and time delay . Furthermore
appropriate oscillators can be dra wn from the reserv oir , such that the period distrib ution
e xhibits multiple peaks 213 , 368 .
Apart from this, learning algorithms for artificial intelligence ranging from generations I-III
featuring units with binary and smooth sigmoidal transfer functions up to spiking units
369
can be tested. The implementation of a XOR logic gate would serv e as a first proof of
concept. An additional benefit of the experiment in comparison to a simulation is the inherent
heterogeneity , noise and aging in the experiment. Such obstacles would need to be ov ercome
in a working real-w orld neural net, too. Especially the role of noise in learning
370
is an
interesting v enue as it could play a constructi ve role as in stochastic
371
and coherence
resonance 372 , 373 .
Changing the chemical BZ reagents appropriately allo ws for experiments with e xcitable
instead of oscillatory units (Fig. C.2 ). This opens the possibility for an e xperimental study
of b ump states
239
, thought to play an important role in short-term memory
374
. Coupling
chemical oscillators together might also allo w for chaotic units and possibly bistable elements.

4.6 Future Directions 83
Spreading e xcitation wa ves are also commonly observ ed in central pattern generators
209
(CPG)
that determine the locomotion of in vertebrate and v ertebrate species, such as lampre y fish
375
and urodele amphibians
376
. Furthermore a CPG has been sho wn to successfully manage
the motion of a robot
211
and adapt to e xternal stimuli. This allows for the de v elopment of
a "chemical brain", consisting of chemical micro-oscillators, that learns and controls the
mo vement of a robot.
From a more general standpoint, it would be interesting to experimental ly verify the curv ature-
induced propagation f ailure together with the accompanying generalized eik onal equation
198
on an network, where the front curv ature κ is replaced by the node degree k : κ → k − 2.
The e xperimental setup also opens the possibility for the synchronization and spreading of
wa ves in netw orks with time-dependent connecti vities, as they are found in social contact
377
and tra v el networks
378
. In the latter cases, excitation wa ves are related to the dissemination
of opinions 379 and the proliferation of infectious pathogens.
Reducing the bead distance on the acrylic glass plate allo ws for experiments with local
dif fusiv e coupling in addition to light-mediated nonlocal or global coupling. The interplay
between both coupling schemes might lead to the emer gence of interesting dynamics, which
might resemble those in neuronal assemblies with nonlocal, biochemical synapses and local,
electrical gap junctions 170 , 380 .

Conclusion
In this thesis a number of self-or ganized patterns, that exhibit spatio-temporally periodic
synchronized acti vity , are elucidated in numerical simulations and chemical experiments.
Special focus is gi ven to the propagation of e xcitation wa ves on dif ferent topologies.
In three spatial dimensions an unperturbed scroll ring far from an y boundaries ordinarily
either contracts and v anishes or undergoes a ne gati ve line tension instability that ends in
W infree turbulence. In chapter 2 we provide e xperimental and numerical e vidence that a
scroll ring with positi ve filament tension in spatial confinement does not only contract, b ut
may also e xpand, which was pre viously associated with ne gati ve line tension e xclusi v ely .
Be yond this unexpected finding, we also observ e that boundary interaction can stabilize the
ring radius for a long time, resulting in an ef fecti v e autonomous pacemaker in a homoge-
neous medium without defects. T o describe the experiments, we de velop a semi-analytical
kinematical model that e xplicitly takes the boundary interaction into account. It succeeds in
accurately reproducing all numerical observ ations and illuminates the underlying mechanism
of boundary-mediated stabilization in the e xperiment
162
. These findings are rele v ant to
pathological scroll wa ves of electrical acti vity in the myocardium
36
, which are naturally
bounded by the spatial e xtent of the heart.
On a small network of relaxation oscillators, an e xcitation wa ve simply spreads from one
node to its neighbors. Surprisingly the wa ve propag ation is also bound by the symmetry
properties of the network as elucidated in chapter 3 . Those wa ves that spread on the netw ork
between symmetry clusters e xclusi vely are found to be the gene ralization of target w a v es
kno wn from continuous activ e media. Both originate from the site of highest frequency in the
system, kno wn as the pacemaker . Their emitted wa ves spread while satisfying all symmetries
of the underlying system, be it Euclidean symmetries or network automorphisms. Utilizing
discrete chemical oscillators and numerical simulations we find that the domain of target
wa ve synchronization in the space of coupling strength and oscillator mismatch resembles
an Arnold tongue
217
. Besides the generalization of target w a ves, these findings might be of

86 Chapter 4
rele vance t o assemblies of nerve cells, that support tra veling cortical e xcitation wa v es
220
and
neural central pattern generators that control rhythmic muscle mo v ement 209 .
The final chapter 4 details our e xperimental and numerical endeav ors regarding the suc-
cessful v erification of spiral wa ve chimeras in a lar ge array of nonlocally coupled chemical
micro-oscillators. T o this end we de v eloped an experimental setup that consists of more than
N = 2500
catalyst-loaded cation-e xchange beads in Belousov-Zhabotinsk y solution forming
a reserv oir of chemical relaxation oscillators. Each oscillator can be monitored via fluores-
cence and independently addressed with light illumination from a spatial light modulator . A
self-b uilt sie ve machine facilitates the homogenization of the oscillator population in order to
perform e xperiments on synchronization. This enabled us to construct a two-dimensional
array of
40 × 40
oscillators to in vestigate the spiral w a v e chimera, predicted by Kuramoto in
2002
271
. Furthermore we elucidated the phase diagram of nonlocally delay-coupled oscilla-
tors on a two-dimensional grid. At the transition of spiral wa ve chimeras to incoherence, we
found that relaxation oscillators e xhibit a unique scenario, which in v olves the splitting of the
incoherent core
334
. Further splitting e vents lead to the formation of a v ortex glass consisting
of chimera cores. In addition to spiral patterns, we also found
d
-cluster states. T aking
into account the measured phase response curv e of type zero
87
we de vised a theoretical
mechanism for the emer gence of clusters in strongly coupled oscillators that we verified in
numerical simulations. Our findings are of significance beyond chemical oscillators, since
the y apply to strongly coupled relaxation oscillators that are found in cardiac and cortical
ensembles
239 , 353 , 359
. Apart from biological rele v ance, nonlocally coupled oscillators also
play a role in physical systems, such as the photoelectrodissolution of doped silicon
248
,
arrays of opto-mechanical oscillators
230
and ultracold atoms
237
as well as metamaterials of
superconducting quantum interference de vices 234 .

A ppendix A
Dimension r eduction of oscillators and
oscillatory patter ns
Detailed dynamical systems for real-world oscillators may contain such a lar ge number
of components, that they are dif ficult to analyze, see for example the MBM model of the
BZ reaction
381
or detailed neural
382
or cardiac cell models
383
. While special methods,
like adiabatic elimination and bath approximation (appendix C.2 ) or principal component
analysis
384
can be applied to remov e a number of components, the most elemental reduction
is possible by e xploiting the topological structure of the limit cycle: A closed circle embedded
in a high-dimensional phase space 87 , 213 , 347 .
The treatment of spatially coupled oscillators falls pre y to the same problem, but further
amplified. Their description is usually gi ven in the framew ork of partial dif ferential equations
(PDE), which are infinite-dimensional due to the additional dependence on a continuous
spatial v ariable. Discretizing a PDE via method of lines
158
yields a finite, b ut still lar ge
dimensional system. In the case of three spatial dimensions, the resulting dimension is:
d = n x × n y × n z × n c
, where
n c
is the number of components of the local dynamics and
the other v ariables giv e the number of cells in each spatial directions. For the scroll ring
dynamics (chapter 2 ), the dimension is on the order of 10 6 .

88 Dimension reduction of oscillators and oscillatory patterns
A.1 Discr ete oscillators
Figur e A.1 | Phase reduction of a limit cycle.
(a) In a planar dynamical system
( C.7 )
a representati v e
limit cycle
Γ
(black) is located around an unstable fixed point (unfilled circle). The velocity along
Γ
is not a constant as indicated by the non-equidistant colored isochrones indicate. The extension of the
isochrones beyond
Γ
is calculated with the asymptotic phase. (b) W ith a suitable transformation, the
non-uniform dynamics on
Γ
can be mapped to uniform motion on a topologically equi v alent circle.
Each point on the circle is parametrized by a phase φ , which follo ws ( A.3 )
Coupled discrete oscillators can be described with a system of coupled ordinary dif ferential
equations:
d
d t c i = f i ( c i ) + K
N
∑
j = 1
I i j ( c i , c j ) . (A.1)
Here the v ectors
c i , c j ∈ R n c
are v ectors of the dynamical v ariables with
n c
components for
each of the
N
coupled oscillators inde xed by
i , j
. The dynamics of the coupled units may
range from harmonic to relaxation oscillations and is gi v en by the local dynamics
f i ∈ R n c
.
The oscillators interact with each other described by the scalar coupling strength
K ∈ R
multiplied with the interaction function
I i j : R n c × R n c → R n c
. The indices of
f i
and
I i j
reflect
the possible heterogeneity of each node. First W infree
12
and later K uramoto
213
e xploited
the fact, that an uncoupled dynamical system,
d
d t c = f ( c ) (A.2)
sho wing time-periodic behavior
c ( t ) = c ( t + T )
must feature a limit c ycle
Γ
embdedded in
a high-dimensional phase space
R n c
that is topologically equi valent to a one-dimensional

A.1 Discrete oscillators 89
ring embedded in two dimensions. While before the dynamic state was defined by the v alue
of the dynamic v ariables
{ c i } i = 1 ,..., n c
, the state along the one-dimensional manifold is gi ven
by the position on the circle or its phase
ϕ
(Fig. A.1 ). Points on the limit cycle are mapped
directly to corresponding phase v alues, such that the phase gro ws linearly with constant
angular frequenc y ω = 2 π / T 0 ∈ R in time 87 , 213
d
d t ϕ = ω (A.3)
= ⇒ ϕ linear ( t ) = ω t . (A.4)
Here, the of fset
ϕ 0
is set to zero. This is also called the temporal phase definition
87
.
Alternati ve phase definitions include the geometrical phase mapping
331
, which takes into
account the spatial position on the limit c ycle:
ϕ geometric ( t ) = arctan ( v ( t ) − v 0
u ( t ) − u 0 ) . (A.5)
Here
( u 0 , v 0 )
is a selectable reference point in phase space and
arctan
is the four -quadrant
in verse tangent. The dif ficulty in applying this method successfully lies in determining a ref-
erence point, that remains inside the modified limit c ycle in the presence of perturbations
385
.
Another method employs the Hilbert transform
H 386
, to extend a single real-v alued input
u ( t )
into a comple x one u ( t ) + i H ( u ( t )) and determine its phase:
H ( u ( t )) = 1
π −
∫ + ∞
− ∞
u ( ˜
t )
t − ˜
t d ˜
t (A.6)
ϕ Hilbert ( t ) = arctan ( H ( u ( t ))
u ( t ) ) . (A.7)
Ho wev er , the wa ve form
u ( t )
needs to be preprocessed, such that its zeroth Fourier mode
v anishes
387
:
u ( t ) − u 0
. Otherwise if
u ( t )
is entirely positi ve, not all phase v alues
[ 0 , 2 π [
are accessible. Since any frequenc y can be mapped from a time-dependent to a constant
frequenc y by reparametrization
ω ( t ) = ω ( t ( s ) ) / d s
d t = const , (A.8)
and due to its simplicity , we will choose ( A.3 ) as the phase definition in what follo ws.
The phase corresponding to points c outside the limit c ycle R 2 \ Γ can be calculated via the
asymptotic phase
ϕ ( c )
. It is defined via the phase corresponding to the point on the limit

90 Dimension reduction of oscillators and oscillatory patterns
Figur e A.2 | Phase definitions and dynamics.
(a) T ime series of dynamical v ariables during an
unperturbed relaxation oscillation with period
T 0
in a model for the BZ reaction
( C.5 )
. Both dy-
namical v ariables correspond to chemical concentrations. (b) Comparison of dif ferent time-phase
mappings: linear (blue), geometric (yellow), Hilbert (green) (c) The impact of perturbations at
phases
ϕ
in terms of resultant phase change
∆ ϕ ( A.9 )
along the limit cycle is quantified as phase
response curves. Here the linear time-phase mapping
( A.4 )
is employed. The perturbations are in the
form of additi ve inputs
u → u + ∆ u
(blue) and
v → v + ∆ v
(yello w). While perturbations in
u
adv ance
the phase by resetting it to zero, perturbations in v delay the phase.
c ycle, that the trajectory reaches for
t → ∞
starting from state
c
. This procedure allo ws for
finding isochrones, lines of equal asymptotic phase ϕ ( c ) in phase space (Fig. A.1 ).
Coupled oscillators continuously perturb each others phases by adv ancing or delaying each
other . The phase shift
∆ ϕ
due to a perturbation is quantified with the phase response
curv e (PRC)
Q ( ϕ )
. It can be calculated directly by measuring the perturbed period
T p
in
relation to the unperturbed period T 0 :
∆ ϕ = 2 π T 0 − T p
T 0 . (A.9)
An e xample for a PRC is sho wn in figure A.2 c for the ZBKE model
( C.5 )
. Negati ve v alues
of
∆ ϕ
stand for phase delays and positi ve v alues for phase adv ances. Note that there are
opposing sign con ventions in the literature for the PRC
388
. In addition the pulse-shape
P ( ϕ )
defines the perturbation magnitude that an oscillator imposes on its neighbors depending
on its current phase
ϕ
. Combining these two aspects and accounting for heterogeneity of
oscillators, leads to the e volution for a single oscillator coupled to its N neighbors:
d
d t ϕ i = ω i + Q i ( N
∑
j = 1
K j P j ( ϕ j ) , ϕ i ) . (A.10)

A.1 Discrete oscillators 91
Analytical progress can be made under the assumption of identical coupling strength
K j = K
,
homogeneous transfer
P j = P
and response
Q i = Q
to perturbations as well as weak interac-
tion, which means that the state ne v er de viates far from the limit c ycle, since the attracting
local dynamics dominates o ver the coupling. This has two important implications: The total
result of all perturbations can be e xpressed as their linear superposition,
Q ( K , ϕ ) = K Q ( ϕ i )
,
and the ef fect of the perturbations ev olve on such a slo w time scale that their ef fect is negligi-
ble during a single oscillation. Instead the perturbations accumulate o v er many periods on a
slo w timescale. T aken together this simplifies equation A.10 to the W infree model 12 :
d
d t ϕ i = ω i + K Q ( ϕ i )
N
∑
j = 1
P ( ϕ j ) . (A.11)
The ef fect of the accumulated perturbations ov er an oscillation cycle period can be accounted
for by a v eraging ov er a period 213 , 221 :
d
d t ϕ i = ω i + K
N
∑
j = 1
1
2 π ∫ 2 π
0 Q ( ϕ i + ˜
ϕ ) P ( ϕ j + ˜
ϕ ) d ˜
ϕ . (A.12)
Note that
ϕ
was replaced here with
˜
ϕ → ϕ
, such that it does not represent the instantaneous
phase
ϕ
an ymore, b ut the accumulated phase change
˜
ϕ
, that slo wly changes ov er many
periods. Choosing the simplest F ourier series term for the
2 π
-periodic phase response curv e
Q ( ϕ ) = − sin ( ϕ )
and a delta-peak as pulse-shape
P ( ϕ ) = 2 π δ ( ϕ )
to counter the inte gral,
model ( A.12 ) becomes
d
d t ϕ i = ω i + K
N
∑
j = 1
sin ( ϕ j − ϕ i ) . (A.13)
This is the paradigmatic K uramoto model
213
, which is frequently employed for studying
synchronization in lar ge networks.
Alternati vely , the same result can be deri v ed via multiple scale analysis
389 – 391
starting from
( A.1 )
and assuming a limit c ycle solution. For this purpose, we e xploit the two time scales of
the problem. Introducing an additional slo w time scale
t ′′ = ε t
with a small scalar parameter
ε ∈ R
besides the original fast one
t ′ = t
. V ariations on the slo w time scale will only become
rele vant after a long time t → t ′′ / ε .

92 Dimension reduction of oscillators and oscillatory patterns
Thus, taking into account both time scales, an appropriate perturbation ansatz for the solu-
tion c i of ( A.1 ) is:
c i ( t ) =
∞
∑
n = 0
ε n C i , n ( t ′ , t ′′ ) ≈ C i , 0 ( t ′ , t ′′ ) + ε C i , 1 ( t ′ , t ′′ ) . (A.14)
The total time deri vati v e of equation A.1 is transformed via the chain rule, such that
d
d t c i ( t ) ≈ ∂ C i , 0
∂ t ′ + ε ∂ C i , 0
∂ t ′′ + ε ∂ C i , 1
∂ t ′ . (A.15)
T o incorporate the assumption of weak coupling, the coupling strength
K
is supposed to be
of order
ε
:
K = ε
. Plugging in the ansatz
( A.14 )
and applying T aylor expansion, the right
hand side of ( A.1 ) in up to first order of ε becomes
f ( c i ) ≈ f ( C i , 0 + ε C i , 1 ) ≈ f ( C i , 0 ) + ε D f ( C i , 0 ) C i , 1 ,
(A.16)
K
N
∑
j = 1
I ( c i , c j ) ≈ ε
N
∑
j = 1
I ( C i , 0 + ε C i , 1 , C j , 0 + ε C j , 1 ) ≈ ε
N
∑
j = 1
I ( C i , 0 , C j , 0 ) , (A.17)
where D f is the Jacobian of the local dynamics f . Overall the resulting equation reads
∂ C i , 0
∂ t ′ + ε ∂ C i , 0
∂ t ′′ + ε ∂ C i , 1
∂ t ′ = f ( C i , 0 ) + ε D f ( C i , 0 ) C i , 1 + ε
N
∑
j = 1
I ( C i , 0 , C j , 0 ) . (A.18)
Collecting terms by order of ε :
ε 0 : ∂ C i , 0
∂ t ′ = f ( C i , 0 ) (A.19)
ε 1 : ∂ C i , 1
∂ t ′ − D f ( C i , 0 ) C i , 1 =
N
∑
j = 1
I ( C i , 0 , C j , 0 ) − ∂ C i , 0
∂ t ′′ (A.20)
As e xpected, the zeroth order
( A.19 )
returns the original dynamical system, whose solution
is the unperturbed limit c ycle:
C i , 0 ( t ′ , t ′′ ) = c i , LC ( t ′ + ϕ i ( t ′′ )) . (A.21)
In the solution the position on the limit c ycle is gi v en by the fast time
t ′
together with a phase
of fset ϕ i ( t ′′ ) , which e volv es on the slo w time scale, b ut not on the fast one.

A.1 Discrete oscillators 93
The e xpression for the first order in
ε ( A.20 )
is in the form of an inhomogeneous linear
dif ferential equation for C i , 1 after insertion of ( A.21 ),
( ∂
∂ t ′ − D f ( c i , LC ) ) C i , 1 =
N
∑
j = 1
I ( c i , LC , c j , LC ) − c ′
i , LC
d ϕ i
d t ′′ . (A.22)
This relation can be formulated as a solv ability condition determined by the Fredholm Alter-
nati ve
392 a
. In Dirac notation
394 , 395
the theorem can be formulated for a linear dif ferential
operator
ˆ
L
with v ectors
| x ⟩
and dual v ectors
⟨ x |
, describing the e xistence of a solution to the
equation
ˆ
L | x ⟩ = | b ⟩ . (A.23)
A solution | x ⟩ to equation A.23 e xists if and only if:
⟨ x | b ⟩ = 0 , (A.24)
where
⟨ x |
is the left eigen vector to the adjoint eigen value problem to the eigen v alue 0, also
called response function:
⟨ x | ˆ
L † = 0 . (A.25)
Simply stated,
( A.23 )
only has a solution if the inhomogeneity
| b ⟩
is orthogonal to the kernel
subspace of the adjoint operator ˆ
L † , which is populated by the response functions ⟨ x | .
Applied to ( A.22 ) the scalar product of ( A.24 ) e valuates to:
1
T ∫ T
0 Q ( t ′ ) · [ N
∑
j = 1
I ( c i , LC ( t ′ ) , c j , LC ( t ′ − ( ϕ i − ϕ j ) )) − c ′
i , LC
d ϕ i
d t ′′ ] d t = 0 , (A.26)
Note that the phase of fset is shifted,
t ′ + ϕ → t ′
, such that oscillator
i
has a relati ve phase of
zero at time
t = 0
. The response function of
( A.22 )
is denoted
Q ( t )
, which is identical to the
phase response curv e in case of weak perturbations
348
. Utilizing the normalization condition
a
For finite v ector spaces, the Fredholm Alternati ve is a corollary of the rank-nullity theorem
393
:
dim A = dim ( ker A ) + dim ( im A )
. If the kernel space is empty then the operator is injecti ve and surjecti ve.
Otherwise it is neither . The latter case applies for v anishing eigen v alues, since
A x = 0 x = 0
. Consequently , a
solution x to A x = b is not guaranteed to exist.

94 Dimension reduction of oscillators and oscillatory patterns
from Malkin’ s theorem 396
1
T ∫ T
0 Q ( t ′ ) · d f ( c i , LC )
d t ′ d t = 1 , (A.27)
it is possible to deri ve the e volution equation for the slo wly changing phase of fset:
d
d t ϕ i = ε 1
T ∫ T
0 Q ( ˜
t ) ·
N
∑
j = 1
I ( c LC ( ˜
t ) , c LC ( ˜
t + ( ϕ j − ϕ i ) )) d ˜
t , (A.28)
= ε ω i + ε
N
∑
j  = i
H ( ϕ j − ϕ i ) . (A.29)
Note that for identical oscillators the interaction function
H
results in the frequenc y de viation
from the unperturbed oscillation:
H ( ϕ i − ϕ i ) = ω i 348
. Replacing the periodic function
H ( ϕ j − ϕ i )
by its first F ourier mode
sin ( ϕ j − ϕ i )
times a scalar factor
K
, it is possible to
reco ver the K uramoto model 213 :
d
d t ϕ i = ω i + K
N
∑
j = 1
sin ( ϕ j − ϕ i ) . (A.30)
The scaling factor
ε
was absorbed into the time v ariable
ε t → t
, such that the dynamics occur
on a slo w time scale in agreement with ( A.13 ).
The synchronization dynamics of the K uramoto model can be readily analyzed with a mean
field approach
213
. The mean field is gi ven by the centroid of the oscillators in the comple x
plane
Re i Ψ = 1
N ∑
k
e i ϕ k with R = ⏐ ⏐ ⏐ ⏐ ⏐
1
N ∑
k
e i ϕ k ⏐ ⏐ ⏐ ⏐ ⏐
. (A.31)
Plugging these e xpressions back into the Kuramoto model
( A.30 )
, leav es us with a mean-field
equation:
d ϕ i
d t = ω i + K R sin ( Ψ − ϕ i ) (A.32)
This equation describes ho w a single oscillator is coupled to the effecti v e remainder of
the network, which is giv en by the Kuramoto order parameter
R
and its phase
Ψ
. This
sho ws that a single oscillator is alw ays attracted to the mean phase of the b ulk, since the
frequenc y decreases due to the coupling if the oscillator is ahead and the frequency increases
in the opposite case. In addition, the more oscillators align, the stronger is the pull of the

A.2 Continuous oscillator fields 95
b ulk on single oscillators. Ho we ver , since
sin ( ϕ )
is bounded, oscillators with a frequency
that e xceeds the maximum pulling
K cr = K / ∆ ω
, can not be synchronized. The frequenc y
dif ference
∆ ω = ω i − ω 0
is taken between the indi vidual
ω i
and b ulk frequency
ω 0
. The
latter is gi ven as the arithmetic mean of the frequenc y distrib ution,
ω 0 = 1
N
N
∑
i = 1
ω i , (A.33)
because the coupling function is odd. At a suf ficiently high coupling strength oscillators
may start firing in unison by under going dif ferent synchronization scenarios, such as quorum
sensing
397
, mobbing
398
, discontinuous e xplosi ve synchronization
367 , 399
or the continuous
K uramoto phase transition 12 , 213 , 313 , 400 , 401 .
A.2 Continuous oscillator fields
In the pre vious section, an oscillator w as perturbed due to its coupling to other discrete
oscillators. F or continuous oscillator fields the interaction is incorporated as a dif fusion term
∆ d c =
d
∑
i = 1
d 2 c
d x 2
i
, (A.34)
which gi ves rise to a partial instead of an ordinary dif ferential equation. In order to reduce
the dimensionality for d = 3 dimensional v orte x dynamics in reaction dif fusion problems,
d
d t c = f ( c ) + D ∆ 3 c , with c = c ( x , y , z , t ) , (A.35)
it is adv antageous to focus on the dynamics of self-organized structures called filaments
402
.
These are lines of phase singularities to which spiral-shaped solutions
c 0 ( r , t ) = c 0 ( r , θ , t )
of the two-dimensional reaction dif fusion equation
d
d t c = f ( c ) + D ∆ 2 c , with c = c ( x , y , t ) , (A.36)
are attached. In a co-rotating frame of polar coordinates,
˜
θ ( t ) = θ − ω t
, the solutions
c 0 ( r )
are stationary and can be determined by solving the equi v alent nonlinear eigen v alue problem
( D ∆ 2 + ω ∂ θ + f ( · ) ) c 0 ( r , θ − ω t ) = 0 . (A.37)

96 Dimension reduction of oscillators and oscillatory patterns
Figur e A.3 | Goldstone modes and r esponse functions of spiral wa ves.
(a) The stationary
u
-
concentration profile of a spiral wa ve simulated with the FHN model
( C.7 )
on a circular domain
enclosed by Neumann boundaries. (b) Real part of the rotational Goldstone mode
| 0 ⟩
. (c) Real part of
the corresponding response function
⟨ 0 |
. It sho ws the spatial regions that are sensiti ve to perturbations,
which will excite the Goldstone mode
| 0 ⟩
. The black contour line is for comparison. Simulation and
mode determination is performed with DXSpiral 406 .
Stability properties can be inferred from the corresponding linearized dif f erential operator
ˆ
L
.
Its eigen value problem for general eigen v ectors | n ⟩ reads:
ˆ
L | n ⟩ = ( D ∆ 2 + ω ∂ θ + D f ( c 0 ) ) | n ⟩ = ( µ + i ω n ) | n ⟩ (A.38)
⟨ n | ˆ
L † = ⟨ n | ( D ∆ 2 − ω ∂ θ − D f ( c 0 ) ) = ⟨ n | ( ν − i ω n ) (A.39)
Here the adjoint eigen value problem
( A.39 )
is included as well. Note that the left
⟨ n |
and
right eigenmodes | m ⟩ fulfill the orthogonality relation ⟨ n | m ⟩ = δ nm .
From
( A.37 )
it is apparent that we ha v e to find the eigen vectors
c 0 ( r )
to the potentially
de generate eigen v alue
λ = 0
, which are called the Goldstone modes
206
. Perturbations in the
form of these modes are shape-preserving and lead solely to translation and rotation of the
spiral wa ve.
Due to the Euclidean symmetries of the infinite plane
339 , 403 , 404 b
, there are three possible
eigenmodes with a v anishing real part of the eigen v alue
µ
: one for rotation
( | 0 ⟩ = ∂ θ c 0 ( r ) )
and two for translations
( |± 1 ⟩ = ( ∂ x ± i ∂ y ) c 0 ( r ) )
. Note that in the non-rotating frame, the
translational modes are the deri vati v es along the Euclidean basis vectors.
T o gain further understanding of the role of response functions
132 , 133 , 407
, let us consider
a small perturbation
ε h ( c , r , t )
on the two-dimensional spiral w a ve field
c 0 ( r , t )
, which
is a stable solution of
( A.37 )
. Since the perturbation is weak, it can only change the
b
Here we limit ourselves to the special Euclidean group SE(2), which only contains rotation and translation
symmetries. Reflection is excluded, since it would flip the chirality of the spiral pattern leading to a violation of
topological char ge conserv ation 38 , 405 .

A.2 Continuous oscillator fields 97
rotation frequenc y
ω = ω 0 + ε ω 1
and induce translational mo vement
v = ε v 1
of the slightly
perturbed concentration field
c = c 0 + ε c 1
. In a frame that follo ws these rotations and
translations, ( A.36 ) together with the perturbation becomes
( ∂ t − ( ω 0 + ε ω 1 ) ∂ θ − ε v 1 • ∇ )( c 0 + ε c 1 )
= f ( c 0 ) + ε D f ( c 0 ) c 1 + D ∆ 2 ( c 0 + ε c 1 ) + ε h (A.40)
On dif ferent scales we collect
ε 0 : ( ∂ t − ω 0 ∂ θ ) c 0 = f ( c 0 ) + D ∆ 2 c 0 (A.41)
ε 1 : ∂ t c 1 − ω 1 ∂ θ c 0 − v • ∇ c 0 = ˆ
L c 1 + h (A.42)
Noting that the stable spiral wa ve solution c 0 is stationary in the co-rotating frame and thus
∂ t c 0 = 0 , ( A.41 ) is identical to ( A.37 ) and determines the unperturbed pattern ( ω , c 0 ) .
Since projections of response functions
⟨ n |
on
c 1
can be absorbed into
c 0
, we can introduce
a gauge condition
407
,
⟨ n | c 1 ⟩ = 0
, such that
⟨ n | ∂ t c 1 ⟩ = 0
and
⟨ n | ˆ
L c 1 ⟩ = 0
. This relationship
simplifies the e valuation of the Fredholm alternati v e of
( A.42 )
, which in volv es the projection
on response functions:
∂ t φ = ε ω 1 = − ε ⟨ 0 | h ⟩ (A.43)
∂ t R x = ε v x = − ε ⟨ + | h ⟩ (A.44)
∂ t R y = ε v y = − ε ⟨−| h ⟩ . (A.45)
These are the equations of motion for a spiral wa ve under weak perturbations, such as a
parameter gradient, advecti ve fields or periodic forcing
408
. For the special case of a parameter
step in an additi ve e xcitability threshold
h ( x , y , t ) = e 1 ∆ pH ( x − x 0 )
, the theory predicts a
stable spiral wa ve drifting in parallel to the step, which was confirmed e xperimentally as
well 362 . The projection can be e v aluated as 132
⟨ n | h ⟩ ∝ ∫∫
R 2
Y n ( r ) • h ( r ) d 2 r , (A.46)
where for clarity constant factors stemming from the transformation between fix ed and
co-rotating frame ha ve been omitted. This sho ws that a perturbation
h
is most ef fectiv e of
inducing motion of the spatially unbounded spiral wa ve pattern
c
, when it has a lar ge ov erlap

98 Dimension reduction of oscillators and oscillatory patterns
Figur e A.4 | Frenet-Serr et frame.
A curved scroll w a ve with a twisted filament (red ribbon)
simulated with the FHN model
( C.7 )
. Initial twist and torsion were initialized from a random walk in
x
and
y
directions as well as rotation phase
ϕ 0
. Frenet-Serret coordinate frame is locally attached to
the curved filament to parametrize space ( A.47 ).
inte gral with any of the response functions
Y n
, which are localized in the neighborhood of
the spiral tip 133 c .
In order to make progress on the three-dimensional problem
( A.35 )
, we adopt from the
dif ferential geometry of space curves the curvilinear Frenet-Serret frame
410
. The filament
curv e can be described by a vector
R ( s )
, which is parametrized by its arclength
s
. T o each
point of the filament, a triad of orthonormal v ectors, the tangent vector
T ( s ) = ∂ s R ( s )
, the
normal v ector
N ( s ) = ∂ ss R ( s )
and the binormal v ector
B ( s ) = T ( s ) × N ( s )
, are attached lo-
cally . In addition this description naturally introduces the torsion
τ ( s ) = − ∂ s B ( s ) · N ( s )
and
curv ature
κ ( s ) = ∂ s T ( s ) · N ( s )
of the filament. Globally , positions in space are parametrized
with
r = r ( s , p , q ) = R ( s ) + p B ( s ) + q N ( s ) . (A.47)
c
Beyond the response functions, the entire spectrum of adjoint eigenmodes are localized at the spiral tip
409
.
They are important in the case of unstable spiral w av es as well as stable, confined spiral wa ves in bounded
domains, where the continuous symmetries of the special Euclidean group SE(2) are broken.

A.2 Continuous oscillator fields 99
Making use of the chain rule and the Frenet-Serret equations, the space and time deri v ati ves in
( A.35 )
can be e xpressed in the Frenet-Serret coordinates (Fig. A.4 )
d
. Under the assumption
that filament quantities v ary on different spatial scales 118 ,
O ( ε − 1 ) : T , (A.48)
O ( ε 0 ) : N , B , c 0 , c 0 , θ , c 0 , θ θ , c 0 , p , c 0 , q , (A.49)
O ( ε 1 ) : τ , φ s , q , p , c 0 , s , (A.50)
O ( ε 2 ) : c 1 , c 0 , ss , R t , N t , B t , φ t , φ ss , τ s , κ , (A.51)
we can again apply multiple scale analysis in conjunction with the Fredholm alterna-
ti ve
( A.24 )
, but no w in space instead of time. Here subscripted v ariables are shorthand
for partial deri vati ves. The perturbation ansatz is gi ven in the co-rotating frame and account-
ing for twist:
c ≈ c 0 + c 1 , with c = c ( s , p , q , t ) = c ( r , θ − φ ( s , t ) − ω t ) . (A.52)
After simplifying
1 − p κ ≈ 1
, since the curv ature is small, and linearizing the local dynam-
ics f , then up to order ε 2 , equation A.35 in the corotating Frenet-Serret frames is:
ˆ
L c 1 ·
· = ( ∂ t − D ∆ 2 − D f ( c 0 ) ) c 1
= f ( c 0 ) + D ( ∂ s ( φ s + τ ) ∂ θ c 0 + ( φ s + τ ) 2 ∂ θ θ c 0 + κ ∂ p c 0 ) + D ∆ 2 c 0
− ( φ t − ω ) ∂ θ c 0 + ( R t + q N t + p B t ) • T ( φ s + τ ) ∂ θ c 0
+ R t • N ∂ p c 0 + R t • B ∂ q c 0 + N t • B ∂ θ c 0 .
(A.53)
This procedure decouples the spiral wa ves in each
p
-
q
plane , since their interaction in
u 1
is on the scale
ε 4
. Introducing the twist
87
,
w = φ s + τ
, for bre vity and collecting terms for
d
Note that the coordinate system is degenerate in the case of a curv e without curv ature. This problem
was remedied in recent w ork by V erschelde et al.
407 , 411
, who replaced the Frenet-Serret frame with the Fermi-
W alker frame, describing trajectories in curv ed space-time in general relati vity
412
, to prov e a geodesic minimal
principle 413 for filament e volution in anisotropic media.

100 Dimension reduction of oscillators and oscillatory patterns
dif ferent powers of ε results in:
ε 0 : 0 = f ( c 0 ) + D ∆ 2 c 0 + ω ∂ θ c 0 , (A.54)
ε 2 : ˆ
L c 1 = w s D ∂ θ c 0 + w 2 D ∂ θ θ c 0 − κ D ∂ p c 0 (A.55)
− φ t ∂ θ c 0 + ( R t + q N t + p B t ) • T w ∂ θ c 0 (A.56)
+ R t • N ∂ p c 0 + R t • B ∂ q c 0 + N t • B ∂ θ c 0 (A.57)
·
· =
| b ⟩ . (A.58)
In the lo west order
( A.54 )
, we recov er
( A.37 )
, the nonlinear eigen value problem that deter -
mines the stationary pulse profile of a spiral wa ve in the corotating frame. F or the equation
in second order we in vok e the Fredholm Alternati v e
( A.24 )
for the rotational
⟨ 0 |
and transla-
tional
⟨±|
response functions. Due to the orthonormality relation between Goldstone modes
and reponse functions,
⟨ n | m ⟩ = δ nm
, a number of terms in the solv ability condition v anish.
Furthermore, Biktashe v sho wed, that products pairing rotational with translational modes and
in volving the dif fusion matrix
D
, e.g.
⟨ + | D 0 ⟩
, a ve rage to zero ov er one rotation period
120
.
The remaining terms re veal the equations of motion for the filament 118 , 120 :
φ t = N t • B + R t • T w + ⟨ 0 | D 0 ⟩ w s + ⟨ 0 | D ∂ θ θ c 0 ⟩ w 2 , (A.59)
R t = ⟨ + | D + ⟩ κ N + ⟨−| D + ⟩ κ B . (A.60)
F or the simple case of a closed filament loop with no twist and constant curv ature, projecting
( A.60 )
into the loop plane, results in the equations for the e volution of ring radius
R
and drift
along the symmetry axis e z :
˙
R = − α / R , (A.61)
˙ z = β / R . (A.62)
Here
α = ⟨ + | D + ⟩ ∈ R
is called the filament tension, since the filament ring will compress
for
α > 0
and e xpand for
α < 0
similar to the elastic properties of a stretched or compressed
rubber -based material. The parameter
β = ⟨−| D + ⟩ ∈ R
gi ves the magnitude of the drift
along the symmetry axis. The ne gati ve sign in the radius dynamics results from the radius
R
mo ving in opposite direction to the normal vector
N
. Note that these equations are only
strictly v alid in the limit of small filament curv ature
κ
and suf ficiently far from system
boundaries. An alternativ e deri v ation arises within a kinematic approach
414
, which is based
on the curv ature induced motion of fronts.

A.2 Continuous oscillator fields 101
In summary , the preceeding machinery allo wed us to reduce a lar ge problem on the order of
10 6
dimensions to a two-dimensional problem amenable to the standard theoretical tools of
planar dynamical systems.

A ppendix B
Implementations
B.1 Experimental setups
This section gi ves a chronological o vervie w of the experimental setups and techniques
de veloped and employed o ver the course of this thesis.
B.1.1 Setup I: Thr ee-dimensional active medium
Spiral wa ves and scroll rings alik e can be created by erasing a part of a planar wa v e
38
.
T o realize excitation w av es in a spatially extended medium, we employ the Belouso v-
Zhabotinsk y (BZ) reaction, which is well-established as a chemical oscillator (appendix C ).
Instead of e xploiting sensiti vities to externally applied fields lik e light (section B.1.2 ), it is
possible to just shake the medium in order to completely erase e xcitation wa ves due to fluid
mixing. Since a part of the original wa ve needs to remain in order to nucleate a scroll ring,
there is a complementary part of the medium that consists of agarose hydrogel. The hydrogel
matrix pre vents hydrodynamic flo ws from occurring and thus preserves the structure of the
e xcitation wa ves. The medium is prepared as follo ws:
A liquid- or gel layer with a height of
4 mm
requires
25 ml
of the follo wing reaction solution:
14 . 5 ml
of bidistilled H
2
O (water),
8 ml 0 . 5 M
H
2
SO
4
(sulfuric acid),
1 ml 1 M
C
3
H
4
O
4
(malonic acid or MA),
1 ml 1 M
NaBrO
3
(sodium bromate),
0 . 5 ml 25 m M
(Fe(o phen)
3
)SO
4
(ferroin) and in case of the gel layer 0 . 2 g (C 12 H 18 O 9 ) n (agarose).
The liquid layer is prepared by adding H
2
O, H
2
SO
4
, C
3
H
4
O
4
and NaBrO
3
to an empty beaker
with a v olume of
50 ml
. Before the solution for the gel layer recei ves the reagents, the ag ar
must be cooked. T o this end, only w ater with agarose and a magnetic stirrer is filled in
another beaker of
50 ml
. This mixture is brought to boil on a heating plate while stirred.
Once the boiling point is reached after about
6
–
7 min
, the beaker is mo ved to an unheated

104 Implementations
Figur e B.1 | Setup I:
Scroll ring formation and time e v olution are monitored spectrophotometrically
in a three-dimensional acti v e medium. White light from a spatially homogeneous light source is
absorbed depending on the oxidation state of the ferroin catalyst. This allo ws for the optical recording
of chemical wa ves with a CCD camera.
plate for the solution to cool for
12
–
13 min
while gently stirred, b ut it is not to solidify .
After this time, H
2
SO
4
, C
3
H
4
O
4
, NaBrO
3
and ferroin are added with a pipette under rapid
stirring. Adding the complete set of BZ reagents initiates the first chemical oscillation, which
is visible as a periodic color change from red to blue. The first oscillation tak es about
2 min
,
after which the solution in the beaker is transferred to a Petri dish. While the gel solidifies,
ferroin is also added to the solution for the liquid layer while stirring strongly . It takes about
5
–
7 min
for solidification to finish, at which point the solution for the liquid layer is added
on top of it in the Petri dish. This completes the preparation of the bipartite activ e medium.
The procedure to generate a scroll ring starts with introducing a silv er wire of high purity
(
99 %
Ag) perpendicular to the gel-liquid interface for
15
–
50 s
. Because the silv er wire
perturbs the local ion balance via formation of AgBr, it initiates a small spherical excitation
wa ve at the point of contact with the silv er wire tip. About
20 s
after its appearance, the
medium is shaken manually to suppress w a v e formation in the liquid layer . Once the spherical
wa ve in the gel layer reaches the desired initial radius
R 0
, shaking is suspended. This allo ws
the wa ve to propagate into the no w calm liquid layer and curl in, which nucleates the scroll
ring. Note that the initial distance to the boundaries
h 1
and
h 2
(Fig. 2.4 ) is determined by the

B.1 Experimental setups 105
used solution v olumes of the gel and liquid layer . Caution has to be taken as to not shak e the
medium for too long, because phase wa ves will emer ge after about
5 min
that start from the
radial Petri dish boundary and tra v el inwards. Their interaction with the spherical wa v e will
disturb the circular shape of the scroll ring and can e ven pre vent it from forming.
T o observe the e volution of the scroll ring, the Petri dish containing the chemical bipartite
medium is placed in a spectrophotometric setup (Fig. B.1 ). The Petri dish has to be placed
perfectly in parallel to the ground to a v oid inclined height gradients in the system. This
is realized by a metal ring that sits on three small scre ws that can be adjusted in height to
correct for inclinations. Before the observ ation begins, the acti ve medium is sealed from
outside oxygen, that is kno wn to negati vely af fect the reaction
415
, by v ery carefully placing a
floating acrylic glass plate on the liquid layer and a plastic lid on the Petri dish. Con vecti ve
temperature e xchange is minimized by attaching a transparent plastic wrap to the bottom of
the metal holding piece under the Petri dish. Furthermore, the entire setup is encased in a
cardboard box.
W ith these precautions in place, the optical observ ation is performed by exploiting the
dif ference in absorption spectra of the catalyst for dif ferent oxidation states. The white light
from a spatially homogeneous light illuminator (Fiber -Lite PL-800) with light b ulb (USHIO
EKE/L
21 V
,
150 W
) gets absorbed strongly by the reduced form of the catalyst, b ut not by
the oxidized form. The largest contrast is obtained for blue w a velengths (Fig. 2.3 c), which
is enhanced by a dichroic filter (Edmund Optics) that is light-transmissi v e for
400
–
500 nm
.
The transmitted light-intensity is recorded in a spatially-resolv ed manner with a CCD camera
(COHU 2122-1000), that relays the data to a frame grabber card (Data T ranslation 3155) to
be sa v ed as 8-bit grayv alue image files for later analysis (Fig. 2.5 ).
B.1.2 Setup II: T w o- and thr ee-dimensional activ e medium
It is possible to make the Belouso v-Zhabotinsky reaction light sensiti ve
416
by utilizing
the photosensiti ve catalyst ruthenium-tris-dimethylene-bip yridine (Ru(dmbipy)
2+
3
)
126
. In
addition the absorption spectra of the reduced, Ru(dmbipy)
2+
3
, and the oxidized form,
Ru(dmbipy)
3+
3
, are suf ficiently dif ferent to allo w for spectrophotometric measurements
(Fig. B.2 ). The oxidized form is approximately transparent to all visible wa velengths, while
the reduced form absorbs blue wa velengths 400 –500 nm (Fig. B.3 a).
T o perturb excitation w av es in a thick three-dimensional medium that absorbs light, large
light intensities are required. These intensities can be achie v ed by tuning the projector (Casio
XJ-A140V). As is sho wn in figure B.3 , the projector nativ ely emits blue, green and red light.
In the e xperiment, we only require blue light. For this reason we remo ve the lenses and
mirrors that inject the red and green beam into the output beam line. Furthermore, there is

106 Implementations
Figur e B.2 | Setup II:
Spectrophotometric setup for spiral wa ve control and scroll ring nucleation
via light. The projector applies spatio-temporally v arying light intensity for control (dark blue arro ws)
of chemical dynamics. The resulting patterns are observ able due the absorption difference in the
oxidation state with light (pale blue arro ws) from a flat LED, which was repurposed from a cell phone.
A hollo w black cardboard box acts as a homogeneous background for the camera recording, because
it does not reflect stray light.
no dedicated source for green light. Instead, blue light is transformed into green light via
a fluorescent layer on the colorwheel. Unfortunately the colorwheel also functions as the
internal clock of the projector , which stops operating without it. The solution is to remo ve
the fluorescent coating and replace the lost mass in the form of copper disks to preserv e
the angular momentum. Another option is to e xtract the colorwheel and connect it to the
projector mainboard from the outside.
Another challenge is the spatial light gradient ov er the area of a projected image. Initial
e xperiments sho wed, that the nati ve heterogeneous light ne gati vely impacts the formation
and lifetime of planar scroll rings.
The spatial light distrib ution of the projector is homogenized by first measuring the nor -
malized sensiti vity at e very pix el of the camera chip
S chip ( x , y )
by taking a snapshot of a
cloud-free blue sk y or the output of an Ulbricht sphere. This information allo ws us to correct
for heterogeneities of the camera chip. Ne xt we assign an image of constant brightness
I 0
as
the projector output. W ith the calibrated camera, we take a snapshot of the emitted projector
light intensity
I hetero ( x , y )
on a fluorescent screen. This allo ws us to compute a pixel-wise

B.1 Experimental setups 107
Figur e B.3 | Control and obser vation by blue light.
(a) Absorption spectra of the reduced (orange)
and oxidized (green) form of the Ru(dmbipy)
2+
3
catalyst. Note that control and observ ation light
can not be spectrally separated as both rely on the absorption peak at
λ = 450 nm
. Thus the light
distrib ution for spatio-temporal control is also visible on the camera image of the chemical pattern.
(b) The projector (Casio XJ-A140V) is modified to emit a peak amount of blue, instead of red or green
light. This is achie v ed by removing the optical elements, such as lenses, mirrors and the colorwheel
(yello w dashed shapes) required for the red and green light to be passed into the main beamline.
multiplicati ve correction filter F correction for later use in experiments:
F correction ( x , y ) = I 0
I hetero ( x , y ) S chip ( x , y ) (B.1)
I corrected ( x , y ) = I hetero ( x , y ) · F correction ( x , y ) . (B.2)

108 Implementations
Figur e B.4 | Light-mediated scroll ring initiation.
(a) W ith the modified spatial light modulator ,
it is possible to emit a spatiotemporal light intensity
I ( x , y , t )
that allo ws for the formation of scroll
rings. It consists of a bright boundary layer , that pre v ents wa v es from the outside to interact with the
scroll ring, and a v ariable light intensity on the inside. (b) Inside the shielded area a scroll ring was
nucleated, that periodically emits circular wa vefronts. The white scale bar corresponds to 1 cm.
Scroll rings are initiated with a spatial light distrib ution
I ( x , y )
consisting of three concentric
elliptic light re gions stacked on top of each other (Fig. B.4 a). The largest circle is v ery
bright and pre vents outside e xcitation wa ves, especially those coming from the hydrogel
boundary , to interfere with the patterns on the inside. The ne xt ellipse is less bright to support
the propagation of e xcitation wa ves. The innermost circle is the darkest. In this region
oscillations may occur spontaneously . Once this oscillation spreads into the e xcitable domain
as an e xcitation wa ve, the innermost dark region is remo v ed to pre vent oscillations from
interfering with the scroll ring. The upper part of the cylindrical e xcitation wa ve in the
e xcitable domain is remov ed with a short b urst of high light intensity . Here, the duration
of the illumination is crucial. If it is too short the cylindrical w a ve is not perturbed, b ut if
it is too long all wa ves are erased. The right duration of illumination finally leads to the
successful nucleation of the scroll ring (Fig. B.4 b).
In contrast to v ersion I (section B.1.1 ), this versatile setup allo ws for creating scroll rings
of lar ge initial radii
R 0
, because interference from the boundaries is suppressed with light.
In addition rings of both chiralities are possible by switching the brightness of the two
inner elliptic re gions. Furthermore an y autocompletion patterns, like spirals
169
or multiple,
interacting scroll wa ves or closed filaments with v arying curv ature, like ellipses, squares
or heart-shaped loops can be realized by starting from a suitable initial condition and then
remo ving the upper half of the wa ve structures with an intense short b urst of light intensity .

B.1 Experimental setups 109
B.1.3 Setup III: Discr ete oscillators
Figur e B.5 | Setup III:
Chemical oscillators in a Petri dish exhibit color changes depending on the
phase in their oscillation cycle. The color information is recorded in grayscale by a CCD camera
mounted under a slight angle to a v oid reflections from the liquid surface. Note that the grayscale
v alue is proportional to the scattered light intensity at bead surface. The grayscale information is fed
back to a computer . The computer calculates the coupling signal
( 3.9 )
, which is projected on each
bead with a spatial light modulator (T oshiba TLP X20). The neutral filter allo ws for reducing the
intensity of the projector without restraining the dynamic illumination range of the projector . The
bandpass filter increases the contrast by blocking irrele v ant wa velengths outside of 440 –460 nm.
This e xperimental setup (Fig. B.5 ) supports experiments on discrete chemical micro-oscillators
that are connected by mutual light perturbations instead of dif fusi ve coupling via mass e x-
change
244 , 245 , 417 – 419
. Discrete chemical oscillators are realized by placing cation-exchange
beads (DO WEX SO W X4-200) soaked with the catalyst ruthenium-tris(bip yridine) (Ru(bipy)
3
)
in a Petri dish filled with BZ solution. Employed reagent concentrations are gi ven in table C.1 .
Since the catalyst is confined to the bead locations, the BZ reaction can only occur at the
bead sites. Each bead is a microporous sphere with a size of about
200 µm
. When loaded
with catalyst and soaked in w ater the diameter increases to about
300 µm
. Resembling a
solidified hydrogel
318
a bead features a tight polymer matrix with ca vities and tunnels for
e xchange of molecules. The polymer matrix consists of long polystyrene chains that are

110 Implementations
crosslinked with di vinylbenzene
420
. It is important to note, that Ru(bipy)
3
is actually able to
slo wly leak from the beads, but at a v ery slow rate. After an hour , there is non-negligible loss
of the catalyst to the surroundings. Hence, the duration of an experiment is limited to about
half an hour , during which oscillation periods change about
10 %
. This issue is resolved
in v ersion V (section B.1.5 ) with the introduction of additional dimethylene ligands in the
comple x catalyst.
As in pre vious experimental setups (sections B.1.1 and B.1.2 ) the chemical oscillation c ycle
can be monitored optically due to changes in the absorption spectra that follo w the oxidation
state of the catalyst. The absorption spectra of the Ru(bipy)
3
catalyst v ery closely resemble
those of Ru(dmbipy)
2+
3
(Fig. B.3 a). In contrast to the other setups, the grayv alues that
are recorded by the CCD camera (Lumenera Infinity 2) are not related to changes in the
transmitted light intensity , b ut are due to light scattered at the bead surfaces. The reason for
this is that the camera vie ws the beads under a small tilted angle to av oid bright reflections of
the projector intensity from the water surf ace. In this reduced state of the catalyst incident
light is absorbed, while in the oxidized state light can pass through the bead and scatter
back from it. This light intensity is measured by the CCD camera as 8-bit grayv alues. T o
enhance the contrast, the measured grayv alues
v i , 0
are normalized by the minimum
v i , min
and
maximum v i , max from a running windo w ov er the last 100 s of the individual timeseries:
v i ( t ) = v i , 0 ( t ) − v i , min
v i , max − v i , min . (B.3)
The resulting normalized grayv alues of each bead are used in an autonomously running
MA TLAB program to calculate the appropriate light perturbations to each oscillator based
on the current state of all of its neighbors in the network
( 3.9 )
. The feedback is applied with
a projector (T oshiba TLP X20) with a dynamic range of 256 distinct light intensity v alues.
Thus the feedback needs to be appropriately rescaled to fit in this range.
Note that the control and measurement light are not spectrally separated as in section B.1.2 .
T o av oid artifacts in the recorded grayv alues, measurement and feedback are performed
alternatingly in a duty c ycle. The measurement duration is essentially gi v en by the exposure
time of the camera, which is about
1 s
. After computing the new feedback v alues, which
only takes a ne gligible amount of time, the feedback is applied for
2 s
. Since the chemical
oscillation period takes
40
–
80 s
, a duty cycle length of
3 s
is too small to induce stroboscopic
artifacts. This was also v erified by v arying the length of the duty cycle, which did not af fect
the e xperimental outcome.
One complication is that the beads are not spatially fixated in the Petri dish. This allows them
to mo ve around. The mov ement is caused by con vecti ve flo ws in the liquid layer due to the

B.1 Experimental setups 111
produced reaction heat as well as the product CO
2
accumulating under the beads until there
is enough to rise up and simultaneously push the bead to a dif ferent location. Experiments
sho wed, that the con vecti ve flo w can be suppressed by reducing the height of the liquid layer
to
2 . 0 mm
. Howe ver , this also worsens aging ef fects due to oxygen inflow through the open
boundary with air .
The solution was to de velop a tracking algorithm, that continuously updated the location
of each bead for the projected image. For each bead, the position of its center of mass is
detected with an image recognition program. The same procedure is repeated for the ne xt
camera snapshot. Then the algorithm compares the ne w positions to the old ones and pairs
up those with the least distance. F or not too lar ge mov ement this algorithm works fla wlessly .
In the case of a collision between two beads the algorithm will f ail. Howe ver , once two beads
collide, the y will also couple dif fusi vely and thus corrupt the e xperiment either way .
B.1.4 Setup IV : Discr ete oscillators
Figur e B.6 | Setup IV :
The projector (Acer P1120) provides optical feedback, that is focused by
lenses at a microfluidic chamber holding BZ droplets on a microscope slide. A camera (QImaging
Retiga 2000R), attached to a microscope (Zeiss Stemi 2000-C) mounted o ver the droplets, records the
chemical oscillation states of the droplets as grayv alues due to changes in absorption spectra of the
catalyst. Based on their arithmetic mean, the intensity of the global light feedback is adjusted.
In this v ariant, BZ droplets
421 , 422
, instead of beads are used as discrete chemical micro-
oscillators in spectrophotometric measurements (Fig. B.6 ). Droplets are micrometer -sized

112 Implementations
Figur e B.7 | BZ Droplets.
(a) The molecular arrangement in the membrane of a lipid monolayer
micelle as positioned with packmol
423
and visualized with VMD
424
. Non-polar heads of the surfactant
(red) are turned to the outside, while polar heads (white) are turned to the inside, where the BZ reaction
(blue) occurs. (b) A hexagonal carpet of droplets as modeled with
( C.6 )
exhibits a T uring pattern
composed of droplets with high and lo w catalyst concentration.
objects with a diameter between
50
–
300 µm
. They contain a small v olume of BZ solution,
which is encased in a monolayer (Fig. B.7 ) of surfactants (Pico-Surf). These lipid vesicles
resemble li ving cells, whose membrane is a lipid bilayer .
Once the BZ droplets are pumped into a rectangular region on a microfluidic chip, the y form
a close-packed he xagonal lattice that is immersed in oil (Fluorinert FC-40). The droplets can
interact with each other , because non-polar chemical compounds can pass through the droplet
membrane and dif fuse through the oil phase. The most significant non-polar intermediate in
the BZ reaction is bromine (Br
2
). That explains why coupling through the oil-phase slows
do wn chemical oscillations in the BZ reaction, because bromine can spontaneously decay to
bromide (Br – ), which acts as the inhibitor in the BZ mechanism (see appendix C.1.3 ).
The BZ solution in the droplets contains a dual catalyst, consisting of ferroin as well as
photosensiti ve Ru(bipy)
3
. This allo ws for the spectral separation of control and observ ation
light. While the oscillation cycle can be manipulated with light of 400 –500 nm (Fig. B.3 a),
it can simultaneously be observ ed at
500
–
600 nm
(Fig. 2.3 c). Ho we ver , an additional
interference filter is also required to e xclude fluorescence (Fig. 4.2 c) of the Ru(bipy)
3
catalyst
at lar ge wa velength λ > 550 nm.
In contrast to v ersion III (section B.1.3 ), care has to be taken in order to not e v aporate the
droplets with illumination from the
200 W
mercury lamp in the projector . For this purpose

B.1 Experimental setups 113
an additional infrared filter , that consists of a water chamber , is included in the beam line
that remo ves heat radiation with λ > 1100 nm.
Note, that droplets can disappear o ver time. This is due to a sudden f ailure of structural
inte grity of the surfactant membrane, which results in the dissolution of the droplet. These
spontaneously appearing defect sites in the re gular hexagonal lattice lead to a drifting motion
of all remaining droplets until the lattice finds a ne w state of minimum ener gy . Furthermore,
the BZ reagents inside the droplets are consumed after about
100
oscillation periods. Since
the membrane is impermeable for polar molecules, it is not possible to replenish them.
The oil phase is only permeable for inhibiting b ut not for acti vating species. As a conse-
quence, T uring patterns pre v alently emerge on he xagonal carpets of BZ droplets (Fig. B.7 b).
Depending on initial conditions the y can take the shape of patches, stripes and spots. Intro-
ducing yet another source of inhibitory coupling via global illumination based on the mean
grayv alue, leads to oscillating antiphase patterns, where direct neighbors ne ver fire together .
In addition, there are also mix ed oscillatory and T uring patterns, where one set of oscillators
ceases their acti vity , while the other oscillators continue to oscillate. Increasing the global
feedback strength enlar ges the domain of the mixed patterns.
B.1.5 Setup V : Discr ete oscillators
Figur e B.8 | Setup V :
A lar ge reserv oir of chemical micro-oscillators is fixated on an acrylic glass
plate in an open thermostatted reactor
126
. During their oscillation cycles the y emit fluorescence
photons (red arro ws), which are recorded as grayv alues with a hardware-triggered (NI-USB 6000)
CMOS camera (Imaging Source DMK 23UX174) and a
50 mm
objecti v e. The v alues are sent to
a computer to determine the projected light intensity
I i
on oscillator
i
according to
( 4.24 )
. This
feedback (blue arro ws) is applied to the oscillators with a spatial light modulator (Casio XJ-A140V).

114 Implementations
V ersion V is specifically de v eloped to support a lar ge number of addressable oscillators and
simultaneously allo w for independent and individual control of e v ery oscillator
334
. The
setup only features a single optical axis due to the fact, that the ruthenium-tris(dimethylene-
bipyridine) catalyst (Ru(dmbip y)
2+
3
) absorbs light around
λ = 450 nm
and emits fluorescence
photons for
λ > 600 nm
. This allo ws for the spectral separation of control and observ ation
light. An additional impro vement o ver v ersion III (section B.1.3 ), is the use of three additional
dimethylene groups as ligands in the ruthenium based catalyst Ru(dmbip y)
2+
3 126
. The y keep
the catalyst sterically fixated in the bead polymer matrix, such that it can not leak. This
greatly reduces aging ef fects. The spatial drifting problem is resolv ed as well by confining
the location of all beads. Every bead is trapped within a well on the acrylic glass plate and
sealed of f under a silica hydrogel layer . The setup also relies on modifying the projector for
high blue light intensities (Fig. B.3 b). Since the cross section of each bead is very small
(Fig. 4.4 a), they require a bright light intensity for successful manipulation. Note that the
setup also allo ws for experiments on continuous systems, as the bead plate can readily be
e xchanged with a plate holding a gel layer . In this case the observ ed grayscale images will be
in verted in comparison to images from v ersion II (section B.1.2 ). This is because the reduced
form of the catalyst emits fluorescence photons, whereas in v ersion II the oxidized form of
the catalyst is transmissi ve for visible light.
Micr o-oscillator pr eparation
It is kno wn, that the size of the beads has an impact on the oscillation period
325
, because
reagent e xchange with the surrounding solution scales with the bead surface area. Here, we
employ a sie ving procedure for bead size homogenization with a custom-b uilt sie v e machine
(Fig. B.9 a). Its components are a default laptop that hosts a frequenc y generator program.
The output is fed to an amplifier (Phase Ev olution RS4) and po wered by a DC generator with
13 V
(Statron, Model 3233). The amplifier is connected to a subw oofer speaker (SXCTR ON,
S8P12W , 12 inch membrane diameter ,
200 W
). Once turned on, the lo w-frequency vibration
of the membrane (
18 Hz
) is transferred to the sie ve to wer that is fixated on the membrane
by two-sided tape and crossed rubber bands. T o reduce mov ement, the speaker is fitted to a
wooden base and weighted with a hea vy lead block. The sie v e to wer consists of a collection
basin, three sie ves of
10 cm
diameter (Retsch) with mesh sizes,
106 µm
,
112 µm
and
125 µm
,
as well as a glass top. W e employ three stages to reduce the load on the sie ve in the middle,
because if there are too man y particles, they clog the sie ve mesh. T o increase the throughput
of the sie ving process, we also add glass beads of
1 mm
diameter (Retsch) to each sie ve. Their
kinetic ener gy pushes the resin through the holes. After two hours of sie ving, the beads are
separated well by their diameters and are collected from the sie v es. The quality of the sie ving

B.1 Experimental setups 115
Figur e B.9 | Size homogenization of micr o-oscillators.
(a) Photograph of the custom-b uilt sie ve
machine. (b) The distrib ution sho ws the narro w period selection used for the e xperiment sho wn in
figure 4.8 . The tails of the distribution are due to oscillator beads with abnormal concentrations of
catalyst and the second peak is due to double occupancy of some wells.
process is controlled by measuring the beads under a basic light field microscope (Motic
SMZ-143 Series) with a mounted camera (ImagingSource DFK 21A UC03). The particles and
their sizes are detected automatically employing OpenCV algorithms in Python. Sie ved resin
beads,
m beads = 1 . 0 g
in
V H 2 O = 5 ml
H
2
O, are mix ed by a v ortex mix er (V elp Scientifica) in
a test tube with
V cat = 15 ml
of
c cat = 1 . 66 m M
Ru(dmbpy)
2+
3
solution. The latter is slo wly
added with a pipette o ver a span of 10 minutes. Mixing continues autonomously for two
days to achie ve a homogeneous catalyst loading of
2 . 5 × 10 − 5
mol Ru(dmbpy)
2+
3
/g resin on
all beads. T o achie v e a desired catalyst load
n load
, the required catalyst concentration
c cat
can
be calculated according to:
c cat [ m M ] = n load [ 10 − 6 molg − 1 ] · m beads [ g ]
V H 2 O + V cat [ ml ] . (B.4)
The outcome of this process is verified by color saturation measurements (Fig. 4.4 b). Note
that an o verloading of beads with Ru(dmbip y)
2+
3
leads to the decay of the catalyst. This
can be inferred from exposure to ultra violet light, to which the intact catalyst responds
with fluorescence. Also the color of beads turns to pitch black, if there was too much
catalyst loaded on them. The catalyst-loaded beads are placed on an acrylic glass plate
with
64 × 44 = 2816
wells. The wells ha ve a depth of
150 µm
, a diameter of
200 µm
, and
a separation of
400 µm
, cov ering an area of
3 . 8 × 2 . 6 cm 2
. T o allo w for distrib ution of the
beads, the plate is coated with methanol or a water -surfactant (T itan X 0.05 %) mixture to

116 Implementations
lo wer the surface tension. A fine brush is used to place a single bead in each well. For
con venient placement, a CCD camera is mounted to a microscope so the placement progress
can be vie wed via a computer screen.
After three hours, the beads are sealed in their wells by applying liquid silica hydrogel with
20
shots from a spray bottle. Surfactant is added to the mixture, to reduce surface tension,
which otherwise induces droplet formation from the sprayed mist. Note that spilling the
liquid hydrogel o v er the plate would pull the beads out of the holes, that the y were carefully
placed in. Afterwards the hydrogel is allo wed to solidify and dry for 30 minutes and the plate
is stored in a mildly acidic solution.
On the ne xt day , the plate, prepared with oscillator beads, is installed in the reactor . Catalyst-
free BZ solution is pumped into the reactor to allo w for the BZ reaction to occur , but only
at the bead locations. From the array of chemical oscillators, we select oscillators based on
their natural oscillation periods, as sho wn in Fig. B.9 (b).
B.1.6 LabVIEW Contr ol Program
Figur e B.10 | Camera display correlation.
This routine projects
5 × 5
crosses subsequently at
equally spaced positions on the reactor plane, that are recorded with the camera. If a cross is not
found, it performs a random walk until it is sharp enough to be detected at a dif ferent location.
The program to control setups II and V is written in LabVIEW as it pro vides an intuiti v e user
interface to the e xperimentalist. Its architecture follo ws an object-oriented approach to allo w
for the simple inte gration of future experiment routines.

B.1 Experimental setups 117
Figur e B.11 | Bead detection and selection.
The image sho ws the bead array on the acrylic glas plate
ov erlaid with red and blue lines based on spatial information that was entered by the e xperimentalist.
Only beads that are located inside blue box es are accepted as v alid oscillators. The scatter plot sho ws
the periods of the currently selected oscillators.
Before an y experiments can be performed, the setup needs to be initialized. It requires a
mapping between coordinates on the virtual projector image and coordinates in the real-
world that are observ ed with the camera. The mapping is used to correctly spatially address
dif ferent locations on a continuous gel or individual nodes in a netw ork. The spatial mapping
is based on a homographic transformation
425
, which corrects for perspecti ve distortions. T o
find it, a number of white crosses are applied by the projector at dif ferent locations on the
reactor plane (Fig. B.10 ). W ith an image recognition routine, these crosses are detected in
the camera images. After all crosses were found, the matrix elements in the homographic
transformation are computed employing a linear re gression.
In the case of e xperiments with networks, the a vail able beads must be located and character-
ized (Fig. B.11 ). Since each bead is located in a well on a rectangular grid, the dimensions
of this grid in the camera image are required by the program to assign each oscillating pix el
in the grid to a bead. Pix els without oscillations of suf ficient amplitude are discarded. Beads
are further filtered based on their size. Even after the mixing procedure there might be fe w
v ery small or very lar ge outliers that are left ov er . Also a single well may be occupied by
two beads, which appears as a single lar ge one. Furthermore the oscillation is tested for its
relaxation character e xhibiting a steep rise and a slo w decay . F alse oscillations can occur that
are due to pinned CO
2
b ubbles periodically gro wing and bursting. Once all true oscillators in
the reserv oir ha v e been identified, they can be further filtered based on the natural oscillation

118 Implementations
Figur e B.12 | Experiment supervision.
This frontpanel allo ws for selecting from a number of
dif ferent types of network e xperiments and initial conditions. While the experiment runs, the li ve
dynamics of a subset of all nodes as well as feedback applied to them is visualized.
periods for e xample to achie v e a narro w or multi-peaked period distrib ution. Another option
is to filter the beads by their responsi veness to light perturbations.
Once all suitable oscillators ha ve been selected, the y can be coupled into a desired network
with a chosen coupling function (Fig. B.12 ). After an initial condition stage during which
indi vidual oscillator phases are entrained by local periodic feedback to sho w a desired
starting condition, the coupling function takes o v er and continues autonomously until the
preprogrammed e xperiment duration expires. Then all data is sa v ed to binary files for later
analysis and visualization. As a means of quickly e xploring parameter space, it is also
possible for the e xperimentalist to change coupling parameters like coupling strength
K
and
time delay
τ
during the e xperiment. The ef fect of these changes can be followed in a li ve
visualization of the oscillators.
V ariations of this program were de veloped in LabVIEW and MA TLAB to suit the require-
ments for e xperiments in the workgroups of K enneth Sho walter in Mor ganto wn and Vladimir
V anag in Kaliningrad.

B.2 Numerical Implementations 119
B.2 Numerical Implementations
Simulations of three-dimensional spatial domains of nonlinear PDEs (chapter 2 ) or nonlocally
coupled equations in two dimensions (chapter 4 ) are computationally v ery expensi ve and
take a long time.
F or this reason a simple, b ut fast forw ard Euler integrator
158 , 426
is implemented. In order to
discretize an autonomous continuous dynamical system,
d
d t c ( t ) = f ( c ( t )) (B.5)
with local dynamics
f
and a dynamical v ariable vector
c ( t )
, the time e volution is discretized
into steps
t → t i = i ∆ t
and the time deri vati ve is replaced with a dif ference quotient. The
shorthand notation c t + 1 = c ( t i + 1 ) is introduced here for clarity ,
d
d t c ( t ) → ∆ c
∆ t = c t + 1 − c t
t i + 1 − t i
= f ( c t ) ← f ( c ( t )) . (B.6)
Solving no w for c t + 1 the forw ard Euler scheme is:
c t + 1 = c t + ∆ t f ( c t ) . (B.7)
Note that the choice of the time index in the local dynamics gi ves rise to either the e xplicit
Euler scheme for c t or the unconditionally stable, implicit Euler scheme for c t + 1 158 .
Introducing spatial coupling leads to a reaction dif fusion equation:
∂
∂ t c ( x , t ) = f ( c ( x , t )) + D ∂ 2
∂ x 2 c ( x , t ) . (B.8)
This PDE is translated into a set of coupled ODEs via the method of lines
158
. Discretizing
space x → x i = i ∆ x and replacing spatial deri vati v es with dif ference quotients yields:
∂
∂ t c i ( t ) = f ( c i ( t )) + D c i + 1 ( t ) − 2 c i ( t ) + c i − 1 ( t )
∆ x 2 . (B.9)
In the final step this system of ODEs can be discretized in the same way as ( B.5 ), yielding:
c t + 1
i = c t
i + ∆ t ( f ( c t
i ) + D c t
i + 1 − 2 c t
i + c t
i − 1
∆ x 2 ) . (B.10)

120 Implementations
Figur e B.13 | Runtime reduction due to GPU algorithm.
(a) Comparison of runtimes on GPU vs
CPU for the three-dimensional dif fusion problem. Note that the tiling algorithm does not lead to an
improv ement, as the stencil size is too small. (b) Runtime comparison for nonlocally coupled systems.
Here, the FHN model is used to portray the speedup for single and double precision. For the tiling
algorithm to be ef fecti ve in the case of double precision numbers, the hardware settings of the GPU,
such as block size of concurrent threads must be tuned finely . For small problems the CPU v ersion
outperforms the GPU version, since it a v oids the ov erhead of transferring data to the GPU. Used
hardware in the test: (CPU) Intel i5-2500 4 cores @ 3.30 GHz, 8 GB RAM; (GPU) Nvidia GTX 970,
CUD A 8.0, block size = 24 2 threads, 4 GB RAM
More accurate higher order methods as well as multi-step methods of the Runge-K utta type,
or predictor -corrector algorithms like Adams-Bashforth
158
, take a longer computation time
a
,
b ut do not lead to qualitati v e dif ferences with the Euler scheme once the time step
∆ t
is small
enough. This is due to the dissipati ve character of the local dynamics, which al ways attracts
the trajectory back to the limit cycle or fix ed points. Note that time-adapti ve methods
158
can
lead to a speed-up for a single oscillator . Ho we ver , for a grid of coupled units, a speed-up
is only achie ved when all oscillators mo ve in unison. In other cases, there is one oscillator ,
which will mo ve in f ast region of phase space and limit the time step to its smallest possible
v alue. This results ef fecti vely in a non-adapti ve method.
Ev en with the simple and fast Euler inte grator , long simulations or parameter searches are
prohibiti vely long. Ho we ver , the numerical problem is v ery amenable to parallelization. The
local dynamics and the coupling of each oscillator can be solv ed consecuti v ely in such a way ,
that the calculations for each oscillator occur mostly independently . F or a lar ge number of
oscillators, the lar ge number of processors on a GPU
428
is perfect. CPU parallelization of 4
cores is already adv antageous but ultimately can not compete with o ver a 1000 parallel cores
of a GPU. F or demanding computations presented in this thesis an Nvidia GTX 970 with 4
GB RAM was used. In the case of three-dimensional scroll rings a speed-up of factor 25 was
a
One exception is the e xponential time-differencing algorithm ETDRK4
427
, which is also very fast for
PDEs in three spatial dimensions. Ho we ver , it is less effecti ve for nonlocally coupled systems.

B.2 Numerical Implementations 121
achie ved and two-dimensional nonlocally c oupled systems ran 2-4 times faster on the GPU
in comparison to the CPU (Fig. B.13 ).
The code for computing spiral wa ve chimeras is a v ailable on public Git repository:
https://github .com/bzjan/Spiral_W a ve_Chimera_Solv er .git
B.2.1 CUD A Solver
F or nonlocal coupling a speed up can be achie ved by e xploiting the shared memory of
the GPU for faster access speeds. Whereas for the diffusion problem only a fe w nearest
neighbors are used for the computation, in the case of nonlocal coupling, a lar ge number of
neighbors will be required. Thus many v alues will be reused multiple times. This fact can
be e xploited by preloading all v alues into the shared memory section of the GPU, which is
kno wn as tiling algorithm 429 .

122 Implementations
Algorithm B.1 |
CUD A pseudocode for solving a reaction-dif fusion equation. CUD A-specific syntax
is highlighted in green. An introduction to the CUD A programming language can be found in the
literature
428 – 430
. Here only a part of loading data from the host to the de vice memory (lines 1-5) is
sho wn as well as the parallelized e valuation of local dynamics (lines 8-18) and coupling (lines 21-42)
together with the function call (lines 45-54).
1 // transfer array to GPU
2 array_size = nx * ny * sizeof ( double2 );
3 cudaMalloc (&(d->c),array_size);
4 cudaMalloc (&(d->cnew),array_size);
5 cudaMemcpy (d->c,h.c,array_size, cudaMemcpyHostToDevice );
6
7
8 // local dynamics (FHN model)
9 __global__ void reaction( double2 * c, double2 * cnew, int len, double dt){
10 double ooeps=1.0/0.05;
11 double a=0.9;
12
13 int i = threadIdx.x + blockDim.x * blockIdx.x ;
14 if (i<len){
15 cnew[i] .x =c[i] .x + dt * ( ooeps * (c[i] .x -1.0/3.0 * c[i] .x * c[i] .x * c[i] .x -c[i] .y ) ); // u
16 cnew[i] .y =c[i] .y + dt * ( c[i] .x + a ); // v
17 }
18 }
19
20
21 // spatial coupling
22 __global__ void diffusion( double2 * c, double2 * cnew, int nx, int ny, const double2 diffs){
23
24 int x = threadIdx.x + blockDim.x * blockIdx.x ;
25 int y = threadIdx.y + blockDim.y * blockIdx.y ;
26 if (x<nx && y<ny){
27 int i d x=x+y * nx;
28
29 int left=idx-1;
30 int right=idx+1;
31 int top = idx + nx;
32 int bottom = idx - nx;
33
34 // Neumann BC
35 if (x==0) left++;
36 if (x==nx-1) right--;
37 if (y == ny-1) top -= nx;
38 if (y == 0) bottom += nx;
39
40 cnew[idx] += diffs * ( c[left] + c[right] + c[top] + c[bottom] - 4.0 * c[idx] );
41 }
42 }
43
44
45 // solve reaction diffusion equation
46 int warpsize=32;
47 dim3 nBlocks((ncomponents * n-1)/warpsize+1);
48 dim3 nThreads(warpsize);
49
50 for ( int t=0; t<nsteps; t++){
51 reaction <<<nBlocks,nThreads>>> (d->c,d->cnew,nx * ny,dt);
52 coupling <<<nBlocks,nThreads>>> (d->c,d->cnew,nx,ny,diffcoeffs);
53 swapGPU(d->c,d->cnew);
54 }

B.2 Numerical Implementations 123
Algorithm B.2 |
CUD A pseudocode for long-range coupling via tiled
429
con v olution. The algorithm
consists of three steps. First, all threads in the block preload data that will later be needed for the
coupling into shared memory for faster access (lines 15-22). The two-dimensional thread indices
in the block are aligned with the positions in the input tile. Howe ver , the output tile is smaller
than the input tile, since edge positions are filled with tile boundary v alues. Thus the con volution
operation (lines 25-36) only uses a subset of the threads a v ailable. In the last part, the result is scaled
appropriately and added on the output array (lines 39-41).
1 template < int maskRadius>
2 __global__ void nonlocal_delay_homo_tiled_zbke2k_2d( double2 * input, double2 * output, \
3 double2 * input_delay, int width, int height, int o_tile_width, \
4 const double * __restrict__ M, const double2 coupleCoeff){
5
6 extern __shared__ double input_shared[];
7
8 int tx = threadIdx.x ;
9 int ty = threadIdx.y ;
10 int col_o = blockIdx.x * o_tile_width+tx;
11 int row_o = blockIdx.y * o_tile_width+ty;
12 int col_i = col_o-maskRadius;
13 int row_i = row_o-maskRadius;
14
15 int idx = col_i+row_i * width;
16 double output_temp{};
17 if ((row_i>=0) && (row_i<height) && (col_i>=0) && (col_i<width)){
18 input_shared[tx+ty * blockDim.x ] = input_delay[idx] .y ;
19 } else {
20 input_shared[tx+ty * blockDim.x ] = 0.0;
21 }
22 __syncthreads() ;
23
24
25 if (ty<o_tile_width && tx < o_tile_width){
26 double ksum{};
27 int maskWidth=2 * maskRadius+1;
28 double input0=input[row_o * width+col_o] .y ;
29 for ( int i=0; i<maskWidth; i++){
30 for ( int j=0; j<maskWidth; j++){
31 output_temp += M[j * maskWidth+i] * !!input_shared[i+tx+ blockDim.x * (j+ty)] * \
32 (input_shared[i+tx+ blockDim.x * (j+ty)] - input0);
33 }}
34 }
35 __syncthreads() ;
36 }
37
38 if (row_o<height && col_o<width && tx<o_tile_width && ty<o_tile_width){
39 output[row_o * width+col_o] += couplecoeff * max(output_temp,-5.25e-4);
40 }
41 }
42
43
44 int maskWidth=2 * maskRadius+1;
45 int o_TileWidth=blockWidth-maskWidth+1;
46 dim3 nBlocks((nx-1)/o_TileWidth+1,(ny-1)/o_TileWidth+1);
47 dim3 nThreads(blockWidth,blockWidth);
48 int mem_size=blockWidth * blockWidth * sizeof ( double );
49
50 nonlocal_delay_tiled_2d<maskRadius> <<<nBlocks,nThreads,mem_size>>> (d->c, \
51 d->cnew,d->cdelay,nx,ny,o_TileWidth,d->mask,d->coupling_coeffs2);
52 }

A ppendix C
Chemistry
C.1 Belouso v-Zhabotinsk y r eaction
The Belouso v-Zhabotinsky (BZ) reaction is the proto ypical chemical oscillator
431 – 434
: Ov er
the course of the reaction it e xhibits periodic changes in the concentrations of intermittently
produced and consumed chemical species.
C.1.1 History & A pplications
Disco vered serendipitously in 1951
43
, it has been employed as an e xperimental proving
ground for a v ariety of counter-intuiti ve mathematically predicted phenomena, such as
deterministic chaos
435 , 436
, chaos control
437
, mix ed-mode oscillations
438
, collecti ve syn-
chronization
397
, spiral wa ve and scroll w a ve dynamics and control
89 , 125 , 439 , 440
, stochas-
tic resonance
441 , 442
, coherence resonance
443 , 444
, T uring mechanism
312 , 445 , 446
and chimera
states 244 , 245 .
In addition a number of applications based on the BZ reaction were de veloped, i.e. chem-
ical diodes
447
, parallelized chemical computation
448 – 451
, memory de vices
452 , 453
, image
se gmentation
454
, quantitati ve chemical sensors
455
, neuromorphic spike-timing dependent
plasticity
25
, autonomously mo ving agents
27 , 117 , 456
, message encryption
457
, biomimetic self-
oscillating hydrogels, whose v olume changes are entrained to the chemical oscillations
23
.
The y open the door for pistons in microfluidic de vices, acti ve metamaterials, peristaltic mass
transport and models of or gans, such as intestine, uterus and the heart, as well as actuators in
soft robotics.

126 Chemistry
C.1.2 A ppar ent violation of the second law of thermodynamics
The reaction was initially dismissed
43 , 431
as impossible, since it apparently violated the sec-
ond la w of thermodynamics:
∆ S ≥ 0
; or equi v alently from a chemical standpoint:
∆ G ≤ 0 458
.
Acceptance was not until after the w ork by Field, Körös and Noyes, who – in elucidating
the underlying chemical mechanism – indeed sho wed that the oscillations did not violate
the second la w of thermodynamics
459 – 463
. The resolution of the paradox is that the oscilla-
tions occur not through the thermodynamic equilibrium b ut far from it. Thus the reaction
enthalpy
G
decreases monotonically while periodically switching between a fast and a slo w
rate. W ithout external replenishment of reagents, the oscillations will e ventually cease and
the system will mo ve to wards thermodynamic equilibrium 33 , 431 .
C.1.3 Reaction mechanism
The Belouso v-Zhabotinsky reaction is the oxidation of an or ganic substrate (usually malonic
acid, MA) in an acidic medium in the presence of a redox-catalyst (M
red
/ M
ox
)
433 , 460
. In the
absence of a catalyst the net reaction,
3 MA + 2 BrO –
3 + 2 H + 2 BrMA + 3 CO 2 + 4 H 2 O , (BZ)
is v ery slo w e v en though its free enthalpy change is v ery lar ge
433
. Introducing a catalyst
into the system allo ws for a faster pathway and leads to the aforementioned concentration
oscillations. The most complete reaction scheme to date in volv es
37
species in 48 elementary
reactions
381
. Ho we ver , the main mechanism
433 , 460
behind the oscillations can be reduced to
a small subset and is depicted in figure C.1 . The oscillation cycle can be di vided into three
distinct phases: A, B and C. In each phase a dif ferent set of chemical reactions takes the
lead o ver the others. The protagonists are bromous acid (HBrO
2
), bromide (Br
–
) and the
oxidized form of the metal catalyst M
ox
. Starting with process B, we find that bromous acid
reproduces autocatalytically
HBrO 2 + BrO –
3 + 3 H + + 2 M ox 2 M red + 2 HBrO 2 + H 2 O , (A UT O)
where the presence of HBrO
2
promotes its o wn further increase. The resulting exponential
gro wth is countered with decay via disproportionation:
2 HBrO 2 HOBr + BrO –
3 + H + . (R7)

C.1 Belouso v-Zhabotinsky reaction 127
Figur e C.1 | Mechanism of the BZ reaction.
On the left the outer circle depicts the most important reactions grouped into 3 processes: A, B
and C. The inner circle sho ws the production and consumption of the most rele v ant chemical species during dif ferent stages of the oscillation
and the accompanying color changes of the reaction solution. At the center of the oscillations lies a sketch of the mechanism: An autocatalytic
species sho ws bistability for certain v alues of a controller species. The controller species grows for lar ge values of the autocatalyst and decays
for small v alues. Thus driving the oscillations. The acronyms in the reactions stand for: malonic acid (MA CH
3
(COOH)
2
), bromomalonic acid
(BrMA BrCH(COOH)
2
) and formic acid (F A HCOOH). On the right, time traces of the rele vant species are obtained by simulations with the
FKN model
( C.1 )
. During the majority of the oscillation cycle, process A dominates. Process B and C take o ver for a short amount of time (see
gray shaded re gions) after the bromide concentration falls belo w a critical v alue. While process A dominates, process C is at work as well, as can
be inferred from the decaying oxidized catalyst concentration.

128 Chemistry
Once both processes balance out, HBrO
2
will reach a quasi-steady concentration. Ho we ver ,
the reaction c ycle does not end here, since the reduced form of the catalyst M
red
in reac-
tion (R6) gains an electron and thus attains its oxidized form M
ox
. Even though M
ox
does
not directly compromise the autocatalytic production, it brings out the seeds for its future
decline by pro viding the fuel for process C.
In its oxidized form, the catalyst M
ox
is an electron donor , also kno wn as an oxidizing agent.
This allo ws M
ox
to oxidize not only MA, the original or ganic substrate, b ut bromomalonic
acid (BrMA) as well:
6 M ox + MA + 2 H 2 O 6 M red + F A + 2 CO 2 + 6 H + , (R9)
4 M ox + BrMA + 2 H 2 O 4 M red + Br – + F A + 2 CO 2 + 5 H + . (R10)
Note that process C is not understood in its entirety
433
, since MA and BrMA gi ve rise to a
lar ge number of dif ferent radicals and or ganic acids. That is the reason why instead of gi ving
a complete set of reactions, figure C.1 only sho ws the rele v ant reactions that are agreed upon
in the literature
433 , 436 , 464
. The most important product of these reactions is bromide Br
–
, the
inhibitor .
Its role becomes clear during process A. In a series of reactions an e xcess Br
–
concentration
leads to the decomposition of dif ferent oxybromine species whose remainders ultimately react
with MA to form ne w BrMA. Among these oxybromine species is the autocatalyst, HBrO 2 ,
as well. Its concentration is kept in a lo w steady state since it is produced, b ut consumed as
well. As long as Br
–
is abo ve a lo w critical concentration
[ Br – ] cr
, HBrO
2
autocatalysis will be
suppressed. Only once enough of Br
–
is consumed during the production of BrMA, Process B
can start ane w and initiate another cycle. Note that the reactions ha ve been renumbered in
comparison to the original FKN paper 460 for clarity .
In summary , process B describes the autocatalytic step or the positi ve feedback loop. Pro-
cess C switches from high HBrO
2
to lo w HBrO
2
by seeding process A with Br
–
which
poisons the autocatalytic step. This completes the delayed negati ve feedback.
From the standpoint of nonlinear dynamics, the reason for the oscillations is very simple. The
core principle of this mechanism can be identified as a switch between two coe xisting, bistable
branches of high and lo w HBrO
2
concentration. At high concentrations of the catalyst or the
controlling species, the transition may only occur from high to lo w HBrO
2
, while at small
concentrations the transition occurs from lo w to high concentrations. This mechanism has
been identified to lie at the heart of chemical oscillations
465
and was successfully emplo yed
to design ne w chemical oscillators 466 , 467 .

C.1 Belouso v-Zhabotinsky reaction 129
C.1.4 Redox catalysts
As is e vident from the oscillation mechanism, a suitable catalyst is required to recei v e
an electron from one species (oxidation, R6) and later donate it to another (reduction,
R9 and R10 ). Thereby forming Br
–
from BrO
–
3
. This kind of catalyst is called a redox-
catalyst
468
. In addition the catalysts for the BZ reaction exhibit dif ferent colors depending on
their oxidation state. This allows for simple visual recording of the concentration oscillations.
Originally Belouso v utilized the transition metal cerium, whose ions are colorless in the
reduced state Ce(III) and appear yello w in the oxidized state Ce(IV) 43 . Another e xample is
the transition metal comple x ferroin, the cation of 1,10-(ortho)-phenanthroline ferrous sulfate
or Fe(phen), which e xhibits a strong contrast between red (Fe(phen)
3+
) and blue (Fe(phen)
4+
)
colors during dif ferent phases of the oscillation cycle
43
. While the different oxidation states
of manganese Mn(II) and Mn(III) also e xhibit color changes between transparent and red,
this catalyst is also employed in setups, where the oscillation state is tracked using magnetic
resonance imaging (MRI)
469
. The oscillation cycle can also be track ed potentiometrically
with concentration-specific electrodes
460
, thermometrically
470
, calorimetrically
471
or via
chemiluminescence 472 , 473 .
Among the BZ catalysts, the ruthenium-tris(2,2’-bipyridine) comple x (Ru(bipy)
2+
3
)) or
Rubipy stands out
434
. Besides allowing for spectrometrical measurements, this catalyst
also introduces the possibility for control, since it is photo-sensiti ve
416
. After absorption of
photons at λ = 452 nm, it turns from a weak reducing agent into a very strong one 452 , 474 ,
Ru(bipy) 2+
3 + ¯
h ω * Ru(bipy) 2+
3 . (R-L1)
The transition to the long-li ved e xcited state
474 , 475
allo ws the redox-catalyst to lend its
electrons e ven more easily to other reagents and thus catalyze more reactions than in the
ground state. It was sho wn that the previously unaccessible reaction,
* Ru(bipy) 2+
3 + Ru(bipy) 2+
3 + BrO –
3 + 3 H + HBrO 2 + 2 Ru(bipy) 3+
3 + H 2 O, (R-L2)
is the main benefactor of the stronger reducing agent
476
. Due to light illumination, the
autocatalytic species HBrO
2
is produced as well as oxidized catalyst Ru(bipy)
3+
3
which leads
to the delayed rise of the inhibitor Br
–
. In summary light illumination has both, a direct
e xcitatory and an indirect inhibitory ef fect on the oscillation cycle.
A later modification of the catalyst adds a dimethylene-group to each bip yridine ligand, which
increases the spectrometric contrast and steric fixation in hydrogel polymer matrices
126
as
well as in cation-e xchange resin particles. Furthermore Ru(bipy)
2+
3
and Ru(dmbipy)
2+
3
emit

130 Chemistry
strong fluorescence intensity
477
at
λ > 600 nm
(Fig. 4.2 c) that allo ws for robust tracking of
the oscillation c ycle (see chapter 4 ). Note that interfering chemiluminescence is suppressed
at lar ge concentration of [H 2 SO 4 ] 478 .
Due to its desirable photoredox properties, the Rubipy catalyst and its deri v ativ es
474
are
popular in fields be yond the Belousov-Zhabotinsk y reaction, such as luminescent sen-
sors
479 , 480
, en vironmentally friendly catalysis
481 , 482
, fuel cells
483
, photo voltaics
484
, light
emitting diodes 485 as well as diagnostic 486 and therapeutic applications 487 .
C.1.5 P arameter drift
Ov er the course of the BZ reaction, the amplitude and the period of oscillations change.
The phenomenon is kno wn as parameter drift or aging
346 , 488
. This introduces a difficulty
absent from idealized numerical simulations. During the chemical reaction, reagents are
con verted into products. The rates of change of the concentrations are functions of the
concentrations themselv es, so they will change o v er time. Howe ver , e v en with an open
system
126
that continuously feeds ne w reagents (BrO
–
3
, MA) into the system and allo ws for
escape of gaseous products, most notably CO
2
, an aging ef fect can still be observ ed. This can
be due to the catalyst slo wly leaking from the system, as in v ersion II (appendix B.1.2 ) and
III (appendix B.1.3 ). After increasing the size of the ligands with three dimethylene groups,
the catalyst molecule is fixated in the hydrogel or bead polymer matrix, which resolves this
particular issue
126
. Ho wev er , e ven with these precautions, there is still a non-v anishing
parameter drift. This might be due to slow oxidati v e degradation of the photocatalyst
489 , 490
or the escape of gaseous intermediates such as Br 2 .
C.1.6 T r oubleshooting
There are a number of strate gies to ov ercome the shortcomings of the BZ reaction. Dif ferent
catalysts can be mix ed resulting in what is called dual catalysts
434
. This way an experiment
can feature the strong optical contrast of the ferroin catalyst with the photosensiti vity of
Rubipy catalysts. Another application is to combine absorption spectra of dif ferent catalysts
for tunable spectrometric properties.
It was sho wn that the rate of parameter drift can be drastically reduced by replacing the
hydrogen-donating acid (H 2 SO 4 or HNO 3 ) with a protic ionic liquid 491 , 492 .
Gaseous CO
2
presents itself as a major problem since it leads to the gro wth of numerous
spherical inclusions that act as defect sites in solidified BZ gel systems. One remedy is to
use c yclohexanedione (CHD) as a replacement for MA as the or ganic substrate, since it does
not lead to CO 2 formation 493 , 494 .

C.1 Belouso v-Zhabotinsky reaction 131
Figur e C.2 | Concentration Space Overview .
Figures sho w the dynamical state of the FKN model
modified to account for light illumination in dependence of initial chemical reagent concentrations of
H
2
SO
4
, NaBrO
3
, MA and the metal catalyst M. States are automatically identified by checking for
monotonous con ver gence to a fixed point and its v alue as well as for oscillations by ev aluating the
dif ference between the maximum and minimum concentration. The black dot is a reference point
for comparison with initial concentrations of
[ H 2 SO 4 ] 0 = 1 . 1 M
,
[ NaBrO 3 ] 0 = 0 . 5 M
,
[ MA ] 0 = 0 . 15 M
and [ M ] 0 = 25 m M .
Mixing the BZ reagent with the correct concentrations does not lead to immediate oscillations.
Usually it takes a fe w minutes up to an hour , before oscillations start. This time interv al is
kno wn as the induction period
461
, during which intermediate species are slo wly b uilt up for
the ensuing oscillations. The induction period can be decreased by adding Br
–
and BrMA to
the initial reagent mixture 326 .
T o ensure reproducibility of experiments it is important to thoroughly remo ve an y trace
amounts of oxybromine species including Br
2
from reactor chambers that are to be used
again. F or one set of experiments, chemicals from only the same batch should be used. In
addition the employed chemical reagents should be of analytical grade as the y can contain
impurities that perturb the reaction
495
. All BZ reagents can be obtained from chemical
v endors. V ariations of the standard reagents, e.g. the Ru(dmbip y)
2+
3
catalyst must be
produced via repeated reflux and recrystallization 126 , 496 , 497 or micro wa v e synthesis 498 .
Finding the right concentrations for an e xcitable or oscillatory system can be accelerated
with a quantitati ve numerical model of the BZ reaction (see FKN model ( C.1 )). Emplo ying
the rate constants compiled in table C.2 , the beha vior of the BZ system can be simulated for
a range of initial reagent concentrations.
The ef fect of the different reagents on the dynamics of the BZ reaction can be seen in
figure C.2 . Increasing the sulfuric acid (H
2
SO
4
) concentration, and thereby the proton (H
+
)
concentration, leads to a transition from a stable fix ed point through a Hopf bifurcation with
consecuti ve Canard e xplosion
499 , 500
to a lar ge stable limit cycle. Follo wing another Hopf

132 Chemistry
bifurcation the fix ed point becomes stable again, b ut this time it is located at lar ge v alues of
the oxidized catalyst. Perturbations to the fix ed point at lo w oxidized catalyst concentration
lead to a lar ge phase space e xcursion, while perturbations to the fixed point at high oxidized
catalyst concentrations do not. So only the fixed point at lo w oxidized catalyst corresponds to
e xcitability . The result of increasing bromate concentration [BrO
–
3
] is the same as increasing
[H
2
SO
4
], while increasing [MA] or the total catalyst concentration has the opposite ef fect.
Note that [H
+
] is not linearly dependent on [H
2
SO
4
], b ut is based on the Hammett acidity
function, whose v alues are gi v en in the literature
501
. These diagrams can be used to tune
concentrations of an e xperiment in order to find the desired dynamical beha vior .
C.1.7 Chemical r ecipes
The concentrations for the dif ferent experiments described in this thesis are listed in table C.1
for reproducibility .
T able C.1 | Employed initial concentrations f or the experiments.
Note that the units of the catalyst
depend on the type of the experiment. In experiments with continuous h ydrogels, the concentration
is gi v en in m M , whereas with discrete particles, the unit is amount of catalyst in moles per mass of
exchange resin in grams.
concentration scroll ring 162 scroll ring 152 netw ork 217 network 334
e xperimental setup v ersion I v ersion II v ersion III v ersion V
[H 2 SO 4 ] 0 ( M ) 0 . 16 0 . 39 0 . 78 0 . 77
[NaBrO 3 ] 0 ( M ) 0 . 04 0 . 2 0 . 48 0 . 51
[NaBr] 0 ( M ) 0 0 . 09 0 . 02 0 . 08
[MA] 0 ( M ) 0 . 04 0 . 17 0 . 08 0 . 16
[M red + M ox ] 0 0 . 5 m M 4 . 5 m M 8 . 3 × 10 − 6 mol/g 2 . 5 × 10 − 5 mol/g
catalyst Fe(phen) 2+
3 Ru(dmbipy) 2+
3 Ru(bipy) 2+
3 Ru(dmbipy) 2+
3
dynamics oscillatory e xcitable oscillatory oscillatory

C.2 Ov ervie w of numerical models 133
C.2 Ov erview of numerical models
The reaction mechanism by Field, Körös and Noyes
460
as detailed abo ve contains 37 chemical
species and 48 reaction steps. It can be translated into a system of first order dif ferential
equations, which are amenable to numerical simulation. The con version in v olves standard
techniques from reaction kinetics
458
: mass action la w , adiabatic elimination, also kno wn as
quasi steady state approximation, and bath approximation, which assumes reactant species to
be so ab undantly a v ailable, that their concentration is constant. The resulting FKN model
consists of se ven coupled ordinary dif ferential equations
460 , 464
, which capture the dynamics
of the BZ reaction semi-quantitati vely:
˙
U = − k 2 U W + k 1 W − 2 k 7 U 2 − k 5 U + k − 5 R 2 + k 6 SR ,
˙
V = − k 9 V − k 10 V + k 6 SR ,
˙
W = − k 2 U W − k 1 W − k 3 W P + k − 3 O + k 4 O + k 10 V ,
˙
O = k 3 W P − k − 3 O − k 4 O ,
˙
P = 2 k 2 U W + k 1 W + k 7 U 2 − k 3 W P + k − 3 O − k 8 P ,
˙
R = 2 k 5 U − 2 k − 5 R 2 − k 6 SR ,
˙
S = k 9 V + k 9 V − k 6 SR .
(C.1)
The v ariables
U
,
V
,
W
,
O
,
P
,
R
,
S
stand for the concentrations of bromous acid (HBrO
2
),
oxidized catalyst (M
ox
), bromide (Br
–
), hypobromous acid (HOBr), bromine (Br
2
), hypobro-
mous acid radical (BrO
2
) and reduced catalyst (M
red
), respecti vely . All
k i
are reaction rate
constants and their v alues are listed in table C.2 . Note that the rate constants and v ariables
ha v e been renamed for consistency throughout this chapter .
T able C.2 | FKN model.
Reaction rate constants of the FKN mechanism and their v alues
421
. Due to
discrepancies in the literature
126 , 326 , 421 , 433
the follo wing rate constants are selected, since the y lead to
agreement with experimental observ ations in the photosensiti v e BZ reaction
464
. The concentrations
of bath reagents are absorbed into the rate constants.
rate constant v alue rate constant v alue
k 1 2 M − 3 s − 1 [H + ] 2
0 [BrO –
3 ] 0 k − 5 2 × 10 8 M − 1 s − 1
k 2 2 × 10 6 M − 2 s − 1 [H + ] 0 k 6 5 × 10 6 M − 1 s − 1
k 3 5 × 10 9 M − 2 s − 1 [H + ] 0 k 7 3 × 10 3 M − 1 s − 1
k − 3 10 s − 1 k 8 9 . 3 M − 1 s − 1 [MA] 0
k 4 29 M − 1 s − 1 [MA] 0 k 9 0 . 05 M − 1 s − 1 [MA] 0
k 5 42 M − 2 s − 1 [H + ] 0 [BrO –
3 ] 0 k 10 1 M − 1 s − 1 [BrMA] 0

134 Chemistry
An analysis of the in volv ed time scales
502
re veals which chemical species may be adiabati-
cally eliminated for further simplification. After taking into account that the total catalyst
concentration is conserv ed and non-dimensionalization, the resulting qualitati ve model reads:
˙ u = 1
ε 1 ( − w ( u − µ ) − u 2 + u ) ,
˙ v = u − v ,
˙ w = 1
ε 2 ( f v + φ − w ( u + µ )) .
(C.2)
The remaining v ariables
u
,
v
,
w
stand for the dimensionless concentrations of HBrO
2
, M
ox
and Br
–
. Con version formulas for the parameters
434
are found in table C.3 . Since the organic
reaction pathways are not kno wn in detail, the production of Br
–
can be estimated with a
stoichiometric factor
f
. This f actor relates the number of produced moles of Br
–
from a mole
of consumed M
ox 433 , 463
. Note that this model is called the Modified Complete Oregonator
415
,
since it incorporates the ef fect of light exposure into the original Ore gonator model
463
. Here,
light illumination was thought to e xclusiv ely produce the inhibitor Br – via
* Ru(bipy) 2+
3 + BrMA Ru(bipy) 3+
3 + Br – . (R-L3)
This reaction is accounted for as an additi ve bromide source term
φ
. Ho we ver , the model
ne glects the excitatory impact of light 476 , as described in reaction R-L2 .
Adiabatically eliminating the inhibitor species
w
, leads to the two-component Ore gonator
model 463 , 502 :
˙ u = 1
ε 1 ( u ( 1 − u ) − u − µ
u + µ ( f v + φ ) ) ,
˙ v = u − v .
(C.3)
T able C.3 | Oregonator models.
Con version formulas
434
for parameters in ( C.2 ) and ( C.3 ). The
same rate constants
464
as in table C.2 are employed. Majuscule and minuscule variables stand for
concentrations with and without dimensions, respecti vely . Bath species are abbre viated as
A = [ BrO –
3 ] 0
and B = [ BrMA ] 0 = 0 . 1 [ MA ] 0 464 and H = [ H + ] 0 (via Hammett acidity function 501 ).
u = 2 k 7
k 5 A U v = k 7 k 10 B
( k 5 A ) 2 V w = k 2
k 5 A W τ = k 10 B t
ε 1 = k 10 B
k 5 A ε 2 = 2 k 7 k 10 B
k 2 k 5 A µ = 2 k 1 k 7
k 2 k 5
k 1 = 2 M − 3 s − 1 H 2 A k 2 = 2 × 10 6 M − 2 s − 1 H k 5 = 42 M − 2 s − 1 H A
k 7 = 3 × 10 3 M − 1 s − 1 k 10 = 1 M − 1 s − 1 B

C.2 Ov ervie w of numerical models 135
T able C.4 | Rovinsk y model.
Rate constants and con version formulas
154
for parameters in
( C.4 )
.
Majuscule and minuscule v ariables stand for concentrations with and without dimensions, respecti vely .
Bath species are abbre viated as
A = [ BrO –
3 ] 0
and
B = [ BrMA ] 0 = 0 . 1 [ MA ] 0 464
,
C = [ M red ] 0 + [ M ox ] 0
and H = [ H + ] 0 (via Hammett acidity function 501 ).
u = 2 k 4
k 1 A U v = 1
C V τ = ( k 1 A ) 2 H
k 4 C t
ε = k 1 A
k 4 C α = k 4 k 8 B
( k 1 AH ) 2 β = 2 k 4 k 13 B
( k 1 A ) 2 H µ = 2 k 4 k 6
k 1 k 5
k 1 = 10 M − 2 s − 1 k 4 = 1 . 7 × 10 3 M − 2 s − 1 k 5 = 10 6 M − 2 s − 1
k 7 = 1 . 5 M − 2 s − 1 k 8 = 2 × 10 − 6 M s − 1 k 13 = 10 − 7 s − 1
Later Ro vinsky and co work ers
154 , 503 , 504
de vised another model ( C.4 ) based on a slightly
augmented FKN scheme that leads to semi-quantitati ve agreement between simulations and
e xperiments in case of the ferroin-catalyzed BZ-reaction:
˙ u = 1
ε ( u ( 1 − u ) − u − µ
u + µ ( β + 2 q α v
1 − v )) ,
˙ v = ( u − α v
1 − v ) .
(C.4)
The v ariables
u
and
v
stand again for the concentrations of HBrO
2
and M
ox
, in this case
Fe(phen)
4+
. Con version formulas for the parameters
154
are found in table C.4 . Note that
the photosensiti vity of the ferroin catalyst
416 , 505
is ne glected. Instead the new parameter
β
accounts for the slo w hydrolysis of BrMA into Br
– 504
. The other important change is the
replacement of the oxidized catalyst
v
with the ratio
α v / 1 − v
. This ratio results from
adiabatic eliminations of or ganic species while taking the sum of reduced and oxized catalyst
concentration as constant.
The ZBKE model by Zhabotinsk y and Epstein
326
and its later refinements by T aylor and
others
244 , 434
b uild on the Ro vinsky model ( C.4 ) b ut in addition take into account e xplicitly
the concentration of the intermediate basic radical of bromous acid (HBrO
+
2
). This species
plays a role in process B during the radicalic oxidation of the catalyst:
BrO 2 + H + HBrO +
2 ,
M red + HBrO +
2 M ox + HBrO 2 ·

136 Chemistry
T able C.5 | ZBKE model.
Rate constants
262
and con version formulas
326
for parameters in
( C.5 )
.
Majuscule and minuscule v ariables stand for concentrations with and without dimensions, respecti vely .
Bath species are abbre viated as
A = [ BrO –
3 ] 0
and
B = [ BrMA ] 0 = 0 . 1 [ MA ] 0 464
,
C = [ M red ] 0 + [ M ox ] 0
and H = [ H + ] 0 (via Hammett acidity function 501 ).
u = 2 k 4
k 5 H A U v = 1
C V τ = ( k 5 H A ) 2
2 k 4 C t µ = 2 k 3 k 4
k 2 k 5 H γ = k − 5
k 6
ε 1 = k 5 H A
2 k 4 ε 2 = ( k 5 H A ) 2
2 k 4 k 6 C 2 ε 3 = k 8
k − 7 H C α = 2 k 4 k 7 k 8 B
k 2
5 k − 7 H 3 A 2 β = 2 k ∗
4 k 9 B
( k 5 H A ) 2
k 2 = 2 . 0 × 10 6 M − 2 s − 1 k 3 = 2 M − 2 s − 1 k 4 = 3 × 10 3 M − 1 s − 1
k 5 = 33 M − 2 s − 1 k − 5 = 4 . 2 × 10 6 M − 1 s − 1 k 6 = 4 . 0 × 10 6 M − 1 s − 1
k 7 = 9 . 2 × 10 − 1 M − 1 s − 1 k 8 / k − 7 = 2 . 5 × 10 − 4 M 2 k 9 = 3 . 3 × 10 − 6 s − 1
Its adiabatic elimination leads to a two-component model in volving the acti v ator
u
([HBrO
2
])
and ox. catalyst v ([Ru(dmbipy) 3+
3 ]):
σ ss = 1
4 γ ε 2 ( √ 16 γ u ε 2 + v 2 − 2 v + 1 + v − 1 ) ,
˙ u = 1
ε 1 ( φ + u ( − 1 − u ) − u − µ
u + µ ( β + q α v
ε 3 + 1 − v ) + γ ε 2 σ 2
ss + ( 1 − v ) σ ss ) ,
˙ v = 2 φ + ( 1 − v ) σ ss − α v
ε 3 + 1 − v .
(C.5)
P arameter formulas
326
are listed in table C.5 . Note that the oxidation of the catalyst
Ru(bipy)
2+
3
by radicals (reaction R6) is irre v ersible
434
in contrast to ferroin. This means
the parameter
δ
in the original model
326
v anishes. For modelling cation-e xchange particles
loaded with BZ catalyst (chapters 3 and 4 ), the ZBKE model turned out to be superior o ver
the Ore gonator
( C.3 )
, because it allo wed for a lar ger spread of periods. In the simulations the
heterogeneity in periods was modeled by the stoichiometric parameter q .
F or BZ nano- and microdroplets
421
, V anag de v eloped a number of models
506 , 507
, which are
based on the original FKN mechanism. The model presented here is designed to explicitly
account for non-polar Br
2
molecules that are e xchanged between the droplets through an oil
phase. T o this end, the disproportionation of Br 2 is taken into account as well,
Br 2 + H 2 O Br – + HOBr + H + ·
In addition the influence of the photosensiti v e catalyst Ru(bipy)
3+
3
is incorporated. The
light interaction is assumed
507
to ha ve an e xclusi ve inhibitory impact via reactions ( R-L1 )

C.2 Ov ervie w of numerical models 137
and ( R-L3 ). Also the decay of the excited state via fluorescent photon emission
328
is taken
into account:
* Ru(bipy) 2+
3 Ru(bipy) 2+
3 + ¯
h ω . (R-L4)
Instead of in voking the bath approximation, the periodic time-e v olution of [BrMA] is e xplicity
accounted for . The number of resulting differential equations are reduced by taking the total
catalyst concentration [M
red
] + [M
ox
] C to be constant and adiabatic eliminations of HBrO ,
HOBr and
*
M
red
. Furthermore terms with the smallest contrib ution to the rate of change
are eliminated
507
. Ho we v er , as in all models presented before, the adiabatic elimination of
HBrO introduces a flaw in model: The autocatalytic production of HBrO
2
(see R6) is not
limited by the concentration of M
red
an ymore. T o resolve this issue, the rate constant
k 5
is
made dependent on M red 506 :
k ′
5 = k 5
C − V
C − V + S min .
Here
S min
is the smallest concentration of reduced catalyst during the oscillation cycle. For
lar ge
V
,
k ′
5 → k 5
b ut for small
V
,
k ′
5 → 0
, which ef fectiv ely stops the autocatalysis. The full
model is:
˙
U = − k 1 U W + k 2 W − 2 k 3 U 2 + k 4 U C − V
C − V + S min ,
˙
V = 2 k 4 U C − V
C − V + S min − k 9 B V − k 10 V + C − V
K L / B + 1 φ ,
˙
W = − 3 k 1 U W − 2 k 2 W − k 3 U 2 + k 7 P + k 9 B V + C − V
K L / B + 1 φ ,
˙
P = 2 k 1 U W + k 2 W + k 3 U 2 − k 7 P ,
˙
B = k 7 P − k 9 B V − k 13 B .
(C.6)
The v ariables
U
,
V
,
W
,
P
,
B
represent the species [HBrO
2
], [M
ox
], [Br
–
], [Br
2
] and [BrMA]
as in the FKN model ( C.1 ). V alues of rate constant are compiled in table C.6 .
From the perspecti ve of nonlinear dynamics, the chemical oscillations in the BZ system
require bistability and a controller species that mov es the system between both stable
branches
465
. The bistability can be realized with a species that has a cubic nullcline. Both
properties are found in the FitzHugh-Nagumo model, which is the paradigmatic model of
neural dynamics 348 :
˙ u = 1
ε ( u − 1
3 u 3 − v ) + φ ,
˙ v = u + a .
(C.7)

138 Chemistry
T able C.6 | V anag model.
Rate constants
507
employed in ( C.6 ). Majuscule and minuscule variables
stand for concentrations with and without dimensions, respecti vely . Bath species are abbre viated as
A = [ BrO –
3 ] 0 , M = [ MA ] 0 , C = [ M red ] 0 + [ M ox ] 0 and H = [ H + ] 0 (via Hammett acidity function 501 ).
S min = √ 3 k r k 10 C / k red
k 1 = 2 × 10 6 M − 1 s − 1 H k 2 = 2 s − 1 H 2 A k 3 = 3 × 10 3 M − 1 s − 1
k 4 = 42 s − 1 H A k 7 = 29 s − 1 M k 9 = 20 M − 1 s − 1
k 10 = 0 . 05 s − 1 M k r = 2 × 10 8 M − 1 s − 1 k red = 5 × 10 6 M − 1 s − 1
K L = 0 . 05
In neural systems,
u
corresponds to the membrane v oltage and
v
to the fast g ating v ariable
432
.
Model
( C.7 )
features a supercritical Hopf bifurcation at
a = 1
with a consecuti ve Canard.
This captures the essential characteristic dynamics observ ed in BZ models (Eqs. C.1 to C.6 ).
The time scale separation in the relaxation oscillations can be tuned via
ε
. The parameter
φ
is introduced additi vely in the acti v ator v ariable
u
to mimic the e xcitatory ef fect of light
illumination in the BZ reaction ( R-L2 ). The dif ferent nature of the models is best illustrated
with a depiction of the phase space (or a projection thereof in the
u
-
v
phase plane) and their
respecti ve phase response curv es (PRC) in figure C.3 .
Quantitati ve agreement between a model and an e xperiment is dif ficult to attain, as not
all v alues of the rate constants are av ailable with high precision
326 , 433
. In addition small
temperature changes will lead to dif ferent rate constants due to the Arrhenius law
458
. Further -
more the purity of the employed reactants is not perfect and may contain impurities, such as
trace amounts of halogens e.g. Br
–
and Cl
–
, which strongly influence the sensiti v e dynamic
beha vior of the BZ reaction 495 .
F or the ferroin-catalyzed BZ reaction, the Rovinsk y model
( C.4 )
ga ve semi-quantitati ve
agreement with the e xperiments on spiral wa ves in continuous media
162
and the FKN
model
( C.1 )
yielded good results for a photosensiti v e BZ system with droplets
464
. The
Ore gonator models,
( C.3 )
and
( C.2 )
, are very useful to qualitati vely describe an e xcitable
BZ system. The latest attempt in elucidating the role of light illumination in the BZ
476
sho wed that the ZBKE model
( C.5 )
incorporates the e xcitatory and inhibitory influence of
light correctly .
Besides the FKN model
461
, e xtensi ve chemical models are a v ailable that take into account a
wide range of concurrent reactions, such as the GTF model
508
, which was later simplified via
principal component analysis
509
. The most recent entry is the MBM model
381
with detailed
dynamics for the or ganic subprocess. The abo v e listing gi ves insight into finding reduced
models of the BZ reaction and an o vervie w of the models used in this thesis.

C.2 Ov ervie w of numerical models 139
Figur e C.3 | Model Over view . All models are compared via their limit cycle, time series and phase
response curve. (a) FKN ( C.1 ), (b) modified complete Oregonator ( C.2 ), (c) Rovinsk y ( C.4 ), (d)
ZBKE, ( C.5 ) (e) FHN ( C.7 ). The unfilled and half-filled markers in the limit c ycle sho w the unstable
nodes and saddle points, respecti v ely . Note that all models show a phase-reset character for species
that are identified as acti vators and a phase-delaying character with lar ge negati ve contrib utions for
inhibitory species.

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