Document [original]

Chaos 30, 033125 (2020); https://doi.org/10.1063/5.0002272 30, 033125

Two populations of coupled quadratic

maps exhibit a plentitude of symmetric and

symmetry broken dynamics

Cite as: Chaos 30, 033125 (2020); https://doi.org/10.1063/5.0002272

Submitted: 23 January 2020 . Accepted: 28 February 2020 . Published Online: 17 March 2020

Ralph G. Andrzejak , Giulia Ruzzene , Eckehard Schöll , and Iryna Omelchenko

ARTICLES YOU MAY BE INTERESTED IN

Collective dynamics of phase-repulsive oscillators solves graph coloring problem

Chaos: An Interdisciplinary Journal of Nonlinear Science 30, 033128 (2020); https://

doi.org/10.1063/1.5127794

A note on exact solutions of the logistic map

Chaos: An Interdisciplinary Journal of Nonlinear Science 30, 033136 (2020); https://

doi.org/10.1063/1.5125097

Two paradigmatic scenarios for inverse stochastic resonance

Chaos: An Interdisciplinary Journal of Nonlinear Science 30, 033123 (2020); https://

doi.org/10.1063/1.5139628

Chaos ARTICLE scitation.org/journal/cha

Two populations of coupled quadratic maps

exhibit a plentitude of symmetric and symmetry

broken dynamics

Cite as: Chaos 30, 033125 (2020); doi: 10.1063/5.0002272

Submitted: 23 January 2020 ·Accepted: 28 February 2020 ·

Published Online: 17 March 2020

View Online

Export Citation

CrossMark

Ralph G. Andrzejak,1,2,a)Giulia Ruzzene,1Eckehard Schöll,3and Iryna Omelchenko3

AFFILIATIONS

1Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138,

08018 Barcelona, Catalonia, Spain

2Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute of Science and Technology, Baldiri Reixac 10-12,

08028 Barcelona, Spain

3Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany

a)Author to whom correspondence should be addressed: ralph.andrzejak@upf.edu

ABSTRACT

We numerically study a network of two identical populations of identical real-valued quadratic maps. Upon variation of the coupling strengths

within and across populations, the network exhibits a rich variety of distinct dynamics. The maps in individual populations can be synchro-

nized or desynchronized. Their temporal evolution can be periodic or aperiodic. Furthermore, one can find blends of synchronized with

desynchronized states and periodic with aperiodic motions. We show symmetric patterns for which both populations have the same type of

dynamics as well as chimera states of a broken symmetry. The network can furthermore show multistability by settling to distinct dynamics

for different realizations of random initial conditions or by switching intermittently between distinct dynamics for the same realization. We

conclude that our system of two populations of a particularly simple map is the most simple system that can show this highly diverse and

complex behavior, which includes but is not limited to chimera states. As an outlook to future studies, we explore the stability of two popu-

lations of quadratic maps with a complex-valued control parameter. We show that bounded and diverging dynamics are separated by fractal

boundaries in the complex plane of this control parameter.

Published under license by AIP Publishing. https://doi.org/10.1063/5.0002272

Chimera states are characterized by the intriguing coexistence

of synchronization and desynchronization in networks. They

were first described in models with a very simple structure,

namely, in ring networks of identical non-locally coupled phase

oscillators,1,2as well as in two populations of identical phase

oscillators with3and without4frequency mismatch between the

populations. Apart from simple phase oscillators,3–7further work

on chimera states in two populations of time-continuous dynam-

ics used phase oscillators with coupling delay,8,9external forcing,9

or with inertia,10–13 Stuart–Landau oscillators,13–15 pulse-coupled

oscillators,16 experimental mechanical17 or chemical oscillators,18

models of neurons,13,19,20 and social agents,21 among others.

Extensive work is also dedicated to chimera states in networks

of time-discrete maps (e.g., Refs. 22–44). There are, however,

only very few studies on chimera states in two populations of

maps.45,46 Here, we address this very simple setting and study

two populations of real-valued quadratic maps. We show that

a plentitude of distinct dynamics can be generated by varying

the coupling strengths within and across the two populations.

After being started with random initial conditions, the maps in

individual populations can synchronize or desynchronize, and

their temporal evolution can become periodic or aperiodic. Fur-

thermore, individual populations can show combinations of syn-

chronized and desynchronized states as well as combinations of

periodic and aperiodic motions. We provide various examples for

which both populations have the same type of dynamics, such as

an aperiodic motion, which is synchronous within populations

but asynchronous across populations. We furthermore show dif-

ferent chimera states, for which the symmetry between the two

identical populations is broken by the network dynamics. For

example, while one population can be synchronous and periodic,

the other population can be asynchronous and aperiodic. A key

Chaos 30, 033125 (2020); doi: 10.1063/5.0002272 30, 033125-1

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

point of this study is that a single and very simple system is

capable of generating highly diverse and complex behavior. It

is this simplicity which allows us to describe the basic mecha-

nism behind the symmetry breaking in our system of all identical

elements.

Lattices of coupled logistic maps have been shown to exhibit

different scenarios of complete or partial synchronization (e.g.,

Refs. 47 and 48). Chimera states in a ring network of nonlo-

cally coupled logistic maps were first described by Omelchenko

and colleagues.22 Nayak and Gupte45 reported on chimera states

in two populations of sine-circle maps. The first experimen-

tal observation of chimeras in electro-optical coupled map lat-

tices in one and two dimensions was provided by Hagerstrom

et al.24 Apart from coupled logistic maps,22,23,27–30,37–39,41–44 chimera

states were illustrated for coupled Henon maps,26,32–34,36 sine-

circle maps,45,46 sine-squared maps,35,44 cosine maps,24 piecewise

linear and logistic maps,25 and cubic maps.31,39,40 They were

found for ring networks with nonlocal coupling,22–26,28–36,39,41–43

ring networks with hierarchical connectivity,38 globally coupled

networks with delay,35 two-dimensional lattices,24,40 as well as

two-,27,34,37 three-,43 and many-layer multiplex networks.39,41 Apart

from phase chimeras,22–24,28–34,36,38–43 the existence of amplitude

chimeras,28–32,34,36,38–41,43 double-well chimeras,31,39,40 nested chimera

states,38 splay chimeras,46 and solitary state chimeras34,36 was shown.

Dynamics were found to show spatial as well as spatiotemporal

chaos (e.g., Refs. 22 and 25). Further work on chimera states in

maps revealed temporal intermittency,29,32 noise-induced inter-layer

switching,44 and relay synchronization.43

By means of a linear change of variables, the logistic map can

be transformed into the quadratic map f(z)=z2+c, which is a par-

ticularly simple map that can show chaos. For complex-valued cand

z, the quadratic map allows one to generate the fractal Julia sets and

Mandelbrot set, which made this map famous, including in popu-

lar science.49 We confine the quadratic map to real values by setting

c= −1.8 and using real-valued initial conditions. Decreasing the

control parameter from c=0.25 to c= −2 leads to a period-

doubling bifurcation scenario. For c= −1.8, the quadratic map

shows chaotic motion, and windows of periodic solutions are close-

by on the axis of the parameter c.

We use a network of two identical populations, Xand Y,

each composed by N=100 identical quadratic maps. Each individ-

ual map is coupled with strength Cwto all other maps within its

population and with strength Cato all maps in the other population,

X:xt+1

i=f(xt

i)+Cw

NXN

j=1[f(xt

j)−f(xt

i)]

+Ca

NXN

j=1[f(yt

j)−f(xt

i)], (1)

Y:yt+1

i=f(yt

i)+Cw

NXN

j=1[f(yt

j)−f(yt

i)]

+Ca

NXN

j=1[f(xt

j)−f(yt

i)]. (2)

Here, tis discrete time in units of the iteration step. We use ran-

dom initial conditions distributed uniformly in [−1.5, 1.5] for the

variables xt=1

iand yt=1

i. They are independent for Xand Yand across

the maps i=1, ...,N.

To characterize the network’s dynamics, we first define the fol-

lowing time-resolved measures. The standard deviation of xt

iacross

nodes (i=1, ...,N) is denoted by σt

xand is used to assess the syn-

chronization within population X. We get σt

x=0, if at time tall

maps are identically synchronized, regardless of whether or not

their motion is periodic. For a non-synchronous motion, we get

σt

x>0, without normalization to some upper bound. Periodicity

with period pis assessed by δt

x,p= h|xt

i−xt+p

i|i, where | · | denotes

the absolute value and the angular brackets indicate averaging across

nodes (i=1, ...,N). A value of δt

x,p=0 means that at time t, all

maps are at the beginning of a period pcycle, regardless of whether

or not they are synchronized across nodes. In general, when we

report the dynamics to be periodic with order p, we imply that pis

the minimum period. Temporal averages of σt

xand δt

x,pare denoted

by σxand δx,p, respectively. The quantities σt

y,σy,δt

y,p, and δy,pare

defined analogously for population Y.50,59 These measures allow us

to characterize different types of symmetry broken chimera states.

A dynamics with broken synchronization symmetry is detected if

σx=0 but σy>0, or σx>0 but σy=0. In contrast, σx=σy=0

or σx≈σy>0 indicates that the synchronization symmetry is not

broken. In an analogous way, the δmeasures assess the periodic-

ity symmetry. The σand δmeasures are independent. For a given

dynamics, the measures can, therefore, determine if both types of

symmetry are broken, only one type of symmetry is broken, or if

the dynamics is symmetric with regard to both synchronization and

periodicity.

Before we explore the two-parameter space spanned by the cou-

pling strengths within populations and across populations, we first

show the results obtained for nine different exemplary combinations

of these coupling strengths. This includes three settings in which the

coupling across populations is stronger than the one within them.

The respective values of Cwand Caare given in Figs. 1–4, where we

show realizations of the dynamics obtained for the nine settings.51

We use the labels 1–9 to refer to the settings in the text and to

identify them in the parameter plane displayed in Fig. 5.

Figure 1(a) shows a realization of the network’s dynamics

obtained for setting 1. Shortly after being started with random initial

conditions, at t≈20, both populations enter into a joint asyn-

chronous and aperiodic dynamics. As a result, we find σt

x≈σt

y>0

[Fig. 1(b)] and δt

x,6 ≈δt

y,6 >0 [Fig. 1(c)]. Starting at t≈50, this

initial symmetry between the two populations’s dynamics is bro-

ken and both pass through some transient motions that are distinct

between them. Eventually, both populations enter into a period-6

motion, Xat t≈100 and Yat t≈120. However, while popula-

tion Yis synchronous across its nodes, Xremains asynchronous.

In consequence, we get δt

x,6 =δt

y,6 =0 and σt

x>0, σt

y=0. This

configuration remains stable, as can be seen from Fig. 1(d), which

shows the same realization of the dynamics but for a later inter-

val. Averaging across this interval, we get σx=0.78, while σy,δx,6,

and δy,6 remain zero. Visual inspection of Fig. 1(d) might suggest

that Xand Yenter into a period-2 and period-3 motion, respec-

tively. However, δt

x,2 and δt

y,3 are nonzero, and p=6 is the lowest

period for which we obtain δt

x,6 =0 and δt

y,6 =0. Accordingly, for

this realization, the network symmetry is broken by the dynamics

Chaos 30, 033125 (2020); doi: 10.1063/5.0002272 30, 033125-2

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

FIG. 1. Periodic dynamics with a broken synchronization symmetry coexists with globally asynchronous and periodic dynamics. Results for two exemplary realizations

obtained for different random initial conditions with setting 1: Cw=2.7 ×10−1and Ca=1.2 ×10−2. (a)–(d) First realization, (e) second realization. Panel (a)–(c) have the

same abscissa. (a) Values of the maps vs discrete time tand index ifor the populations Xand Y. (b) Synchronization measures σt

x(blue) and σt

y(red) for the same interval

as shown in panel (a). These measures take zero values for full synchronization of the corresponding population. (c) Same as panel (b), but for the periodicity measures δt

x,6

(blue) and δt

y,6 (red), which take zero values for period-6 dynamics. (d) Same realization like in panel (a), but for a later time interval. (e) Same as panel (d), but showing the

second realization of the dynamics obtained for setting 1. We used N=100 and c= −1.8.

of its two populations with regard to their synchronization. In

contrast, the symmetry is maintained with regard to the popula-

tions’ periodicity. This dynamics can be regarded as analogous to

phase chimeras previously described in ring networks of coupled

maps.22–24,28–34,36,38–43 Figure 1(e) shows the network dynamics still

for setting 1, but after being started with a different set of ran-

dom initial conditions. In contrast to the first realization discussed

above [Figs. 1(a)–1(d)], for this second realization, both populations

show the same type of behavior. They are periodic (δx,2 =δy,2 =0)

and asynchronous (σx=σy=0.79). Hence, for this second realiza-

tion, the network symmetry is not broken by the dynamics of its

populations. Such a coexistence of qualitatively different dynamics

for different random initial conditions is not a peculiarity of set-

ting 1. We found it for many of the cases we studied. This effect

of initial conditions should be investigated further in future studies

using methods such as those used in Refs. 52–54. In the following

examples, we restrict ourselves to one realization for each setting,

each selected to illustrate a different type of dynamics.

Figure 2(a) displays a realization for setting 2. During approx-

imately the first 80 iterations, both populations behave qualitatively

similar. An increasing number of nodes engages to a motion, which

is almost periodic in time with p=6 and synchronous across nodes.

Chaos 30, 033125 (2020); doi: 10.1063/5.0002272 30, 033125-3

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

FIG. 2. Dynamics with broken synchronization symmetry and broken periodicity symmetry. Panels (a)–(c) are analogous to Figs. 1(a)–1(c), but here for an exemplary

realization of setting 2: Cw=1.7 ×10−4and Ca=1.7 ×10−2with N=100 and c= −1.8.

As a result, σt

x,σt

y,δt

x,6, and δt

y,6 all decrease during this initial phase

[Figs. 2(b) and 2(c)]. Furthermore, this emerging pattern is locked

across the two populations. Subsequently, in population Y, more

and more nodes join, and eventually all nodes are synchronized.

For t>220, we, therefore, get σt

y=0 and δt

y,6 ≈0.01. The contrary

behavior is observed for population X. From t≈80 to t≈150, all

nodes return to an asynchronous and aperiodic motion. The values

of σt

xand δt

x,6 approach the ones they had for random initial con-

ditions at t=1. After these initial transients, population Xevolves

as asynchronous and aperiodic, while Yis synchronous and almost

periodic. Population Ycannot settle to a fully periodic dynamics due

to the aperiodic input from population X, but the deviations from a

period-6 motion remain very small, δy,6 =0.01. Accordingly, in this

example, the symmetry is broken with regard to both the synchro-

nization and periodicity of the two populations. It is analogous to

amplitude chimeras in ring networks of coupled maps.28–32,34,36,38–41,43

Above, we already included a dynamics in which the two

populations behave qualitatively the same, namely, periodic and

asynchronous [second realization of setting 1, see again Fig. 1(e)].

Further symmetric dynamics are shown in Fig. 3. Setting 3 allows

generating aperiodic and asynchronous dynamics [Fig. 3(a)]. Ape-

riodic dynamics, which are synchronous within populations but

asynchronous across the populations, can be obtained with setting

4 [Fig. 3(b)]. Setting 5 results in aperiodic and globally synchronous

dynamics [Fig. 3(c)]. A realization obtained for setting 6 is shown

in Fig. 3(d). Both populations are synchronous and periodic with

p=3. While at first sight, there seems to be a lag-1 synchroniza-

tion between Xand Y, the map values are, in fact, different for the

two populations. This is, however, difficult to perceive at the reso-

lution of the gray scale for their display. Setting 7 yields an example

for a blended pattern resulting in an intermediate degree of peri-

odicity and synchronization [Fig. 3(e)]. This pattern resembles the

initial transient of setting 2 (see again Fig. 2), but remains stable for

this realization of setting 7. At first sight, the realization obtained for

setting 8 [Fig. 3(f)] looks like a periodic and globally synchronous

dynamics. Indeed, all maps are in a period-6 motion. However, two

maps in Xand three maps in Yare detached from their respective

population and are synchronized to each other instead. Accordingly,

this dynamics represents solitary states (cf. Refs. 34,36, and 55).

Furthermore, a closer inspection of the values reveals that neither

the motion of the detached maps nor the motion of the remaining

majority of maps is synchronized across Xand Y. As last example, we

show a realization of the dynamics obtained for setting 9 [Fig. 4]. In

contrast to the examples we described above, this dynamics does not

settle to a stable configuration. After initial transients, population

Xsynchronizes at t≈750, while Yremains asynchronous (σt

x=0,

σt

y>0). Furthermore, Yremains aperiodic (δt

y,12 >0). In contrast,

population Xswitches intermittently between an aperiodic motion

(e.g., δt

x,3 >0 between 1480 <t<1623) and an almost periodic

motion (e.g., δt

x,3 ≈0 between 1624 <t<2513). Such intermittent

behavior was described for chimeras in ring networks of maps29,32

and two populations of phase oscillators with inertia11,12 before.

Chaos 30, 033125 (2020); doi: 10.1063/5.0002272 30, 033125-4

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

FIG. 3. A variety of dynamics without symmetry breaking. All panels display values of the maps vs discrete time tand index ifor the populations Xand Y. Panel (a):

Setting 3. Cw=3.8 ×10−4and Ca=9.5 ×10−5. We get σx=σy=1.1, δx,12 =1.0, and δy,12 =1.1. For such aperiodic dynamics, we report δx,12 and δx,12, since this

includes p∈ {2, 3, 4, 6, 12}. Panel (b): Setting 4. Cw=4.7 ×10−1,Ca=5.9 ×10−5,σx=σy=0, δx,12 =1.1, and δy,12 =0.69. Panel (c): Setting 5. Cw=2.0 ×10−1,

Ca=2.5 ×10−1,σx=σy=0, and δx,12 =δy,12 =1.3. Panel (d): Setting 6. Cw=4.9 ×10−1,Ca=7.8 ×10−3,σx=σy=0, and δx,3 =δy,3 =0. Panel (e): Setting 7.

Cw=1.3 ×10−2,Ca=6.3 ×10−3,σx=0.70, σy=0.66, δx,6 =0.33, and δy,6 =0.32. Panel (f): Setting 8. Cw=3.2 ×10−1,Ca=2.9 ×10−5,σx=0.22, σy=0.18,

and δx,3 =δy,3 =0. In all panels: N=100 and c= −1.8.

In order to describe the mechanism behind our results, we write

the network equations [Eqs. (1) and (2)] in a different form (cf.

Refs. 23,28,30,33,38,40,41,43, and 45). Defining γ=1−Cw−

Ca,F(Xt)=1

NPN

j=1f(xt

j), and F(Yt)=1

NPN

j=1f(yt

j), we get

X:xt+1

i=γf(xt

i)+CwF(Xt)+CaF(Yt), (3)

Y:yt+1

i=γf(yt

i)+CwF(Yt)+CaF(Xt), (4)

where f(z)=z2+ccontinues to be the quadratic map. When

regarded in isolation, and for the ranges of Cwand Caused here,

the rescaled map γfexhibits a period-doubling bifurcation scenario

very similar to the one of f. However, the fine structure of the bifur-

cation diagram and the positions of the bifurcation points with

Chaos 30, 033125 (2020); doi: 10.1063/5.0002272 30, 033125-5

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

FIG. 4. Intermittent switching between different dynamics. Panels (a) and (b) are analogous to the graphics in Fig. 3, but here for two intervals of one realization of setting 9

(Cw=3.7 ×10−5,Ca=1.9 ×10−2,N=100, and c= −1.8). (c) Synchronization measures σt

x(blue) and σt

y(red) for a longer interval, including the intervals of panels

(a) and (b). (d) Same as panel (c), but for the periodicity measures δt

x,6 (blue) and δt

y,6 (red).

regard to the parameter cdepend on γ(results not shown). This

already leads to an impact of the coupling strengths Cwand Ca

on the characteristics of the dynamics. What is more important is

the input by F(Xt)and F(Yt). At the level of individual maps, they

add to the period-doubling bifurcation parameter c. Therefore, even

smallest differences in these terms can push the map from a peri-

odic to a chaotic regime or vice versa. The fact that these terms are

time-dependent increases the complexity further. At the level of the

network dynamics, the F(Xt)and F(Yt)terms break the symmetry

between the two populations since their positions are crossed over

in Eqs. (3) and (4). Therefore, once Xand Yenter into different

types of dynamics, also F(Xt)and F(Yt)differ, resulting in a differ-

ent input for Xand Y. Like reasoned above, these different inputs

can potentially push Xand Yinto different dynamical regimes, again

leading to different F(Xt)and F(Yt), and so on. This process may

then either amplify or reduce the differences in the dynamics of X

and Y. For certain values of Cwand Cain combination with some

initial conditions, this mechanism leads to the stabilization of a

symmetry broken dynamics [see again first realization of setting 1 in

Figs. 1(a)–1(d)]. For the same Cwand Ca, but in combination with

other transitory dynamics, the difference in Xand Ycan fade out.

Accordingly, an initial symmetry breaking between Xand Ydoes not

always lead to the stabilization of a symmetry broken chimera state.

Instead, the system can likewise settle to a symmetric dynamics [see

again second realization of setting 1 in Fig. 1(e)]. For other Cwand

Ca, only symmetry broken (see again setting 2 in Fig. 2) or only sym-

metric patterns are stable [see again settings 3–7 in Figs. 3(a)–3(e).

Setting 8 shows a plentitude of qualitatively different patterns for dif-

ferent initial conditions, only one of which is displayed in Fig. 3(f)].

Finally, again for other combinations of the coupling strengths, the

dynamics might not settle down to a stable pattern but switch inter-

mittently between different dynamical regimes (see again setting 9

in Fig. 4).

After inspecting details of individual realizations for exemplary

pairings of Cwand Ca, we now explore the full two-parameter plane

spanned by these coupling strengths. Figure 5(a) allows identifying

Chaos 30, 033125 (2020); doi: 10.1063/5.0002272 30, 033125-6

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

FIG. 5. Synchronization (a) and periodicity (c) of individual populations as well as the symmetry breaking of these properties across populations [(b), (d), and (e)] in

dependence on the coupling strengths. We vary the coupling within populations Cwand across populations Caequidistantly spaced in 366 steps on a logarithmic scale but

keep N=100 and c= −1.8. The labels 1–9 mark the combinations of Cwand Caused for the nine settings discussed in the text and used as examples in Figs. 1–4.

They are in the same positions in all panels, and their different colors are only used to enhance readability. The cyan diagonal marks Cw=Ca. For each combination, we

generate 50 independent realizations of the network dynamics [Eqs. (1) and (2)] for 50 different random initial conditions, where the same 50 realizations are used across all

panels. For each realization, we then compute the temporal averages σx,σy,δx,p, and δy,pfor t∈[10 000, 12 000]. In all panels, the following maximal and minimal values

are taken across the 50 realizations. (a) Zero values of min{σx}indicate that Xexhibits a fully synchronous motion for at least one of the realizations. (b) At the nonzero

max{|σx−σy|} values, the symmetry between the populations is broken with regard to their synchronization for at least one realization. If, for a given combination of Cw

and Ca, a certain realization results in the minimal value in panel (a), this does neither imply nor rule out that the same realization leads to the maximum value in panel (b).

The reason is that panel (b) does not only depend on σxbut also on σy. The panels (c) and (d) are analogous to (a) and (b), but based on the measures of periodicity δx,12

and δy,12. (e) Nonzero max{|σx−σy| · |δx,12 −δy,12|} values indicate that the symmetry was broken for both synchronization and periodicity in at least one realization. If,

for a given combination of Cwand Ca, a certain realization results in the maximum value in panel (b), this does neither imply nor rule out that the same realization leads to

the maximum value in panel (e). Both |σx−σy|and |δx,12 −δy,12|, as opposed to only one of these differences, have to be high to reach the maxima displayed in panel (e).

Analogous relations hold for panels (d) and (e).

the regions in this parameter plane for which Xshows a fully syn-

chronous motion for at least one of 50 independent realizations

(dark brown). These are complemented by regions for which we

obtain only dynamics with various degrees of desynchronization

(red, yellow to white). The strongest degrees of synchronization

symmetry breaking between the populations [yellow regions in

Fig. 5(b)] are found for a subset of parameter settings at which

individual populations can show fully synchronous dynamics [dark

brown regions in Fig. 5(a)]. A certain symmetry across the diagonals

in the results in Figs. 5(a) and 5(b) can be attributed to the influ-

ence of γ, which is symmetric in Cwand Ca. In contrast, the driving

caused by F(Xt)and F(Yt)in Eqs. (3) and (4) is not symmetric in

Cwand Ca. Somewhat more complicated patterns are seen above

the diagonals in Figs. 5(a) and 5(b), i.e., where the coupling across

populations is stronger than the one within populations. The panels

Figs. 5(c) and 5(d) are analogous to Figs. 5(a) and 5(b), but based

on the measures of periodicity. We observed dynamics with periods

p∈2, 3, 4, 6, 12, and here depict results for p=12, because this

includes all lower order periods. With regard to periodicity, more

complex patterns, including two periodic but not symmetry bro-

ken windows centered around Cw≈4.7 ×10−1,Ca≈4.7 ×10−3,

and Cw≈3.6 ×10−1,Ca≈1.2 ×10−1are seen below the diagonals.

Chaos 30, 033125 (2020); doi: 10.1063/5.0002272 30, 033125-7

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

FIG. 6. Octopus’s garden. In panels (a)–(c), gray colors show the complement of the Mandelbrot set, i.e., the set of complex cfor which the values of an individual quadratic

map initiated at z=0 diverge to infinity. Darker gray indicates a faster divergence. Turquoise areas inside the Mandelbrot set show the set of complex cfor which the

iterates of the two populations of coupled maps [Eqs. (1) and (2)] remain bounded for all times. Dark brown to yellow colors are used for those cfor which these iterates

diverge to infinity, where yellow colors indicate a slower divergence. Panels (a)–(h) are obtained for Cw=2.7 ×10−1and Ca=1.2 ×10−2(setting 1). Panels (b)–(h) are

successive zooms, magnifying the dashed squares in the preceding panels (a)–(g). Axes labels indicate cvalues in the outermost corners. Panels (i)–(l) are obtained for

Cw=1.7 ×10−4and Ca=1.7 ×10−2(setting 2). Here, the instabilities can collide, cross, and overlay each other. In each panel, we use different color codes to enhance

the contrast. Initial conditions are always generated in the following way. We start the uncoupled quadratic map with z=0 and take the i-th and i−1th iterate as initial

condition for the ith map in Xand Y, respectively (i=1, ...,N=100). To allow to better resolve the details of these patterns, all panels are magnified in Figs. 1–12 in the

supplementary material.

Chaos 30, 033125 (2020); doi: 10.1063/5.0002272 30, 033125-8

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

The most prominent region of periodicity symmetry breaking, on

the other hand, is found above the diagonal and coincides with a

region of synchronization symmetry breaking (the region including

settings 2 and 9 and limited from below by setting 7). That does not

necessarily mean that the symmetry was broken in both properties.

Likewise, the symmetry could be broken for the synchronization

in some realizations and for the periodicity in some other realiza-

tions. However, as revealed by panel Fig. 5(e), this region indeed

comprises realizations of the dynamics for which the symmetry is

broken for both synchronization and periodicity. At the diagonals

of the plots, we have Cw=Ca, corresponding to a network with one

population of identical and identically coupled maps. The maximal

across-population differences of the synchronization and periodic-

ity measures [Figs. 5(b),5(d), and 5(e)] can nonetheless be non-zero

since the values for individual populations are non-zero and show

finite-size fluctuations across the two populations. As a whole, the

results shown in Fig. 5, complement the findings first described by

Omelchenko et al.22,23 and Hagerstrom et al.24 according to which

chimera states can be found at the transition between spatial coher-

ence and incoherence in rings and lattices of coupled maps. We

here find chimera states at the transition between synchronous vs

asynchronous and periodic vs aperiodic motions.

In summary, we show that two populations of coupled

quadratic maps show a plentitude of symmetric dynamics and sym-

metry broken chimera states. These patterns arise spontaneously

from random initial conditions, i.e., there is no need to craft initial

conditions to induce them. Abrams et al.4argued that their system

of two populations of coupled phase oscillators was the simplest sys-

tem that supports chimera states. Since our system is based on a

time-discrete map, and moreover on a particularly simple map, we

can claim that our system is even simpler than the one of Abrams

and colleagues. In fact, the results which we present here are sup-

posedly only the tip of the iceberg of the system’s complexity. First

of all, we only show selected realizations for nine different com-

binations of the coupling strengths within and across population.

We observed further types of symmetric and asymmetric patterns,

which are not shown here for the sake of conciseness. We assume

that yet unobserved patterns can be found upon further exploration

of the parameter space spanned not only by the coupling strengths

Cwand Cabut also by the system size Nand control parameter

of individual maps c. This might also include dynamics, which we

could not find despite actively searching for them, such as dynamics

for which the symmetry is broken with regard to their periodicity

but not with regard to their synchronization. Furthermore, we only

studied a limited number of iterations. The dynamics we illustrate

here are typically stabilized after some hundreds of iterations. How-

ever, we cannot rule out that some of these dynamics are, in fact,

long transients that eventually collapse to some other dynamics (cf.

Refs. 11,12,29,32, and 56).

Finally, so far, we have only used a real-valued cfor the

quadratic map f(z)=z2+c. It remains to be explored what types

of dynamics can be discovered for complex values of c. The Mandel-

brot set is defined by the set of complex cfor which the values of the

quadratic map initiated at z=0 remain bounded for all times. For

those cthat form the complement of the Mandelbrot set, the map

values diverge to infinity.49 As an outlook, we present an analysis

(Fig. 6) of the complex values of cfrom within the Mandelbrot set.

We determine for which of these cthe values of our two populations

of coupled quadratic maps remain bounded. This approach is dif-

ferent from ring networks of Nlocally coupled quadratic maps for

which the complex control parameters are varied independently for

each map yielding a spatial Mandelbrot set of dimensionality 2N.57,58

Since we use the same control parameter for all maps, we can display

the results in the complex plane. We find that the cvalues for which

the dynamics remain bounded and those for which the dynamics

escapes to infinity are separated by fractal boundaries fragmenting

the Mandelbrot set in a rich variety of ways (Fig. 6). Evidently, our

system inherits this capacity to generate a fractal structure from

the quadratic map. The high dimensionality of our system, how-

ever, leads to additional complexity, and a study of these intriguing

fractal, self-similar patterns, which at some scales resemble octopus

tentacles, remains the subject for future work.

SUPPLEMENTARY MATERIAL

See the supplementary material consisting of Figs. 1–12 for

enlarged displays of the 12 panels of Fig. 6.

ACKNOWLEDGMENTS

We acknowledge funding from the Spanish Ministry of Econ-

omy and Competitiveness under Grant No. FIS2014-54177-R

(R.G.A. and G.R.), the CERCA Programme of the Generalitat de

Catalunya (R.G.), and the Deutsche Forschungsgemeinschaft (DFG)

under Project No. 163436311-SFB 910 (E.S. and I.O.). We are grate-

ful to Christian Rummel for useful discussions on the manuscript.

REFERENCES

1Y. Kuramoto and D. Battogtokh, “Coexistence of coherence and incoherence

in nonlocally coupled phase oscillators,” Nonlinear Phenom. Complex Syst. 5,

380–385 (2002).

2D. M. Abrams and S. H. Strogatz, “Chimera states for coupled oscillators,” Phys.

Rev. Lett. 93, 174102 (2004).

3E. Montbrió, J. Kurths, and B. Blasius, “Synchronization of two interacting

populations of oscillators,” Phys. Rev. E 70, 056125 (2004).

4D. Abrams, R. Mirollo, S. Strogatz, and D. Wiley, “Solvable model for chimera

states of coupled oscillators,” Phys. Rev. Lett. 101, 084103 (2008).

5A. Buscarino, M. Frasca, L. V. Gambuzza, and P. Hövel, “Chimera states in time-

varying complex networks,” Phys. Rev. E 91, 022817 (2015).

6E. A. Martens, C. Bick, and M. J. Panaggio, “Chimera states in two populations

with heterogeneous phase-lag,” Chaos 26, 094819 (2016).

7M. J. Panaggio, D. M. Abrams, P. Ashwin, and C. R. Laing, “Chimera states in

networks of phase oscillators: The case of two small populations,” Phys. Rev. E 93,

012218 (2016).

8J. H. Sheeba, V. K. Chandrasekar, and M. Lakshmanan, “Globally clustered

chimera states in delay-coupled populations,” Phys. Rev. E 79, 055203 (2009).

9R. Ma, J. Wang, and Z. Liu, “Robust features of chimera states and the imple-

mentation of alternating chimera states,” Europhys. Lett. 91, 40006 (2010).

10T. Bountis, V. G. Kanas, J. Hizanidis, and A. Bezerianos, “Chimera states in a

two-population network of coupled pendulum-like elements,” Eur. Phys. J. Spec.

Top. 223, 721 (2014).

11S. Olmi, “Chimera states in coupled Kuramoto oscillators with inertia,” Chaos

25, 123125 (2015).

12S. Olmi, E. A. Martens, S. Thutupalli, and A. Torcini, “Intermittent chaotic

chimeras for coupled rotators,” Phys. Rev. E 92, 030901 (2015).

13C. R. Laing, “Dynamics and stability of chimera states in two coupled popula-

tions of oscillators,” Phys. Rev. E 100, 042211 (2019).

Chaos 30, 033125 (2020); doi: 10.1063/5.0002272 30, 033125-9

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

14C. R. Laing, “Chimeras in networks of planar oscillators,” Phys. Rev. E 81,

066221 (2010).

15K. Premalatha, V. Chandrasekar, M. Senthilvelan, and M. Lakshmanan,

“Chimeralike states in two distinct groups of identical populations of coupled

Stuart-Landau oscillators,” Phys. Rev. E 95, 022208 (2017).

16D. Pazó and E. Montbrió, “Low-dimensional dynamics of populations of pulse-

coupled oscillators,” Phys. Rev. X 4, 011009 (2014).

17E. A. Martens, S. Thutupalli, A. Fourrière, and O. Hallatschek, “Chimera

states in mechanical oscillator networks,” Proc. Natl. Acad. Sci. U.S.A. 110,

10563–10567 (2013).

18M. R. Tinsley, S. Nkomo, and K. Showalter, “Chimera and phase-cluster states

in populations of coupled chemical oscillators,” Nat. Phys. 8, 662–665 (2012).

19S. Olmi, A. Politi, and A. Torcini, “Collective chaos in pulse-coupled neural

networks,” Europhys. Lett. 92, 60007 (2010).

20T. A. Glaze, S. Lewis, and S. Bahar, “Chimera states in a Hodgkin-Huxley model

of thermally sensitive neurons,” Chaos 26, 083119 (2016).

21J. C. González-Avella, M. G. Cosenza, and M. S. Miguel, “Localized coherence

in two interacting populations of social agents,” Physica A 399, 24 (2014).

22I. Omelchenko, Y. Maistrenko, P. Hövel, and E. Schöll, “Loss of coherence

in dynamical networks: Spatial chaos and chimera states,” Phys. Rev. Lett. 106,

234102 (2011).

23I. Omelchenko, B. Riemenschneider, P. Hövel, Y. Maistrenko, and E. Schöll,

“Transition from spatial coherence to incoherence in coupled chaotic systems,”

Phys. Rev. E 85, 026212 (2012).

24A. M. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and

E. Schöll, “Experimental observation of chimeras in coupled-map lattices,” Nat.

Phys. 8, 658–661 (2012).

25D. Dudkowski, Y. Maistrenko, and T. Kapitaniak, “Different types of chimera

states: An interplay between spatial and dynamical chaos,” Phys. Rev. E 90, 032920

(2014).

26N. Semenova, A. Zakharova, E. Schöll, and V. Anishchenko, “Does hyperbolic-

ity impede emergence of chimera states in networks of nonlocally coupled chaotic

oscillators?,” Europhys. Lett. 112, 40002 (2015).

27S. Ghosh, A. Kumar, A. Zakharova, and S. Jalan, “Birth and death of chimera:

Interplay of delay and multiplexing,” Europhys. Lett. 115, 60005 (2016).

28S. Bogomolov, G. Strelkova, E. Schöll, and V. Anishchenko, “Amplitude and

phase chimeras in an ensemble of chaotic oscillators,” Tech. Phys. Lett. 42,

765–768 (2016).

29T. E. Vadivasova, G. I. Strelkova, S. A. Bogomolov, and V. S. Anishchenko,

“Correlation analysis of the coherence-incoherence transition in a ring of non-

locally coupled logistic maps,” Chaos 26, 093108 (2016).

30S. A. Bogomolov, A. V. Slepnev, G. I. Strelkova, E. Schöll, and V. S.

Anishchenko, “Mechanisms of appearance of amplitude and phase chimera states

in ensembles of nonlocally coupled chaotic systems,” Commun. Nonlinear Sci.

Numer. Simul. 43, 25–36 (2017).

31I. A. Shepelev, A. V. Bukh, G. I. Strelkova, T. E. Vadivasova, and V. S.

Anishchenko, “Chimera states in ensembles of bistable elements with regular and

chaotic dynamics,” Nonlinear Dyn. 90, 2317–2330 (2017).

32N. Semenova, G. Strelkova, V. Anishchenko, and A. Zakharova, “Tempo-

ral intermittency and the lifetime of chimera states in ensembles of nonlocally

coupled chaotic oscillators,” Chaos 27, 061102 (2017).

33N. I. Semenova, E. V. Rybalova, G. I. Strelkova, and V. S. Anishchenko,

“Coherence-incoherence transition in ensembles of nonlocally coupled chaotic

oscillators with nonhyperbolic and hyperbolic attractors,” Regul. Chaotic Dyn.

22, 148–162 (2017).

34A. Bukh, E. Rybalova, N. Semenova, G. Strelkova, and V. Anishchenko, “New

type of chimera and mutual synchronization of spatiotemporal structures in two

coupled ensembles of nonlocally interacting chaotic maps,” Chaos 27, 111102

(2017).

35J. D. Hart, D. C. Schmadel, T. E. Murphy, and R. Roy, “Experiments with

arbitrary networks in time-multiplexed delay systems,” Chaos 27, 121103 (2017).

36E. V. Rybalova, G. I. Strelkova, and V. S. Anishchenko, “Mechanism of realizing

a solitary state chimera in a ring of nonlocally coupled chaotic maps,” Chaos

Solitons Fractals 115, 300 (2018).

37S. Ghosh, A. Zakharova, and S. Jalan, “Non-identical multiplexing promotes

chimera states,” Chaos Solitons Fractals 106, 56–60 (2018).

38A. zur Bonsen, I. Omelchenko, A. Zakharova, and E. Schöll, “Chimera states in

networks of logistic maps with hierarchical connectivities,” Eur. Phys. J. B 91, 65

(2018).

39G. I. Strelkova, T. E. Vadivasova, and V. S. Anishchenko, “Synchronization of

chimera states in a network of many unidirectionally coupled layers of discrete

maps,” Regul. Chaotic Dyn. 23, 948–960 (2018).

40I. A. Shepelev, A. Bukh, T. E. Vadivasova, V. S. Anishchenko, and A. Zakharova,

“Double-well chimeras in 2D lattice of chaotic bistable elements,” Commun.

Nonlinear Sci. Numer. Simul. 54, 50–61 (2018).

41E. Rybalova, T. Vadivasova, G. Strelkova, V. Anishchenko, and A. Zakharova,

“Forced synchronization of a multilayer heterogeneous network of chaotic maps

in the chimera state mode,” Chaos 29, 033134 (2019).

42P. Chandran, R. Gopal, V. Chandrasekar, and N. Athavan, “Chimera states in

coupled logistic maps with additional weak nonlocal topology,” Chaos 29, 053125

(2019).

43M. Winkler, J. Sawicki, I. Omelchenko, A. Zakharova, V. Anishchenko, and

E. Schöll, “Relay synchronization in multiplex networks of discrete maps,”

Europhys. Lett. 126, 50004 (2019).

44Y. Zhang, Z. G. Nicolaou, J. D. Hart, R. Roy, and A. E. Motter, “Critical

switching in globally attractive chimeras,” arXiv:1911.07871 (2019).

45C. R. Nayak and N. Gupte, “Chimera states in coupled sine-circle map lattices,”

AIP Conf. Proc. 1339, 172–180 (2011).

46J. Singha and N. Gupte, “Spatial splay states and splay chimera states in coupled

map lattices,” Phys. Rev. E 94, 052204 (2016).

47K. Kaneko, “Clustering, coding, switching, hierarchical ordering, and control in

a network of chaotic elements,” Physica D 41, 137 (1990).

48K. Kaneko and I. Tsuda, “Chaotic itinerancy,” Chaos 13, 926 (2003).

49H.-O. Peitgen, H. Jürgens, and D. Saupe, Chaos and Fractals: New Frontiers of

Science (Springer Science & Business Media, 2006).

50The σand δmeasures are based on differences of values of the quadratic map,

which are of the order 100. Accordingly, values of these measures on the order of

10−14–10−16 are just above the resolution of the double floating point precision.59

We, therefore, assume that such nonzero but very small values arise from the

limited floating point precision and report them to be zero. Any value ≥10−13

is reported as nonzero with two significant digits.

51We specify the Cwand Cavalues with two significant digits. Due to the chaotic

nature of the dynamics, one needs the exact values and initial conditions at double

floating point precision to reproduce our results. These will be provided along

with the source codes to run our simulations on the institutional web site of the

first author.

52E. A. Martens, M. J. Panaggio, and D. M. Abrams, “Basins of attraction for

chimera states,” New J. Phys. 18, 022002 (2016).

53S. Rakshit, B. K. Bera, M. Perc, and D. Ghosh, “Basin stability for chimera

states,” Sci. Rep. 7, 1–12 (2017).

54Z. Faghani, Z. Arab, F. Parastesh, S. Jafari, M. Perc, and M. Slavinec, “Effects of

different initial conditions on the emergence of chimera states,” Chaos Solitons

Fractals 114, 306–311 (2018).

55Y. Maistrenko, B. Penkovsky, and M. Rosenblum, “Solitary state at the edge of

synchrony in ensembles with attractive and repulsive interactions,” Phys. Rev. E

89, 060901 (2014).

56O. E. Omel’chenko, M. Wolfrum, and Y. L. Maistrenko, “Chimera states as

chaotic spatiotemporal patterns,” Phys. Rev. E 81, 065201 (2010).

57P. Liu, “Control and synchronization of Mandelbrot sets in coupled map

lattice,” Nonlinear Dyn. 73, 299–310 (2013).

58D. Wang, S. Liu, Y. Zhao, and C. Jiang, “Control of the spatial Mandelbrot set

generated in coupled map lattice,” Nonlinear Dyn. 84, 1795–1803 (2016).

59IEEE Standards Committee, “754-2008 IEEE standard for floating-point arith-

metic,” IEEE Comput. Soc. Std 2008, 1–70.

Chaos 30, 033125 (2020); doi: 10.1063/5.0002272 30, 033125-10

Published under license by AIP Publishing.