Chaos 25, 083104 (2015); https://doi.org/10.1063/1.4927829 25, 083104
© 2015 AIP Publishing LLC.
Nonlinearity of local dynamics promotes
multi-chimeras
Cite as: Chaos 25, 083104 (2015); https://doi.org/10.1063/1.4927829
Submitted: 12 March 2015 . Accepted: 22 July 2015 . Published Online: 06 August 2015
Iryna Omelchenko, Anna Zakharova, Philipp Hövel, Julien Siebert, and Eckehard Schöll
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Nonlinearity of local dynamics promotes multi-chimeras
Iryna Omelchenko,
1,a)
Anna Zakharova,
1,b)
Philipp H€
ovel,
1,2,c)
Julien Siebert,
1,d)
and Eckehard Sch€
oll
1,e)
1
Institut f€
ur Theoretische Physik, Technische Universit€
at Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
2
Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universit€
at zu Berlin, Philippstraße 13,
10115 Berlin, Germany
(Received 12 March 2015; accepted 22 July 2015; published online 6 August 2015)
Chimera states are complex spatio-temporal patterns in which domains of synchronous and asyn-
chronous dynamics coexist in coupled systems of oscillators. We examine how the character of
the individual elements influences chimera states by studying networks of nonlocally coupled
Van der Pol oscillators. Varying the bifurcation parameter of the Van der Pol system, we can
interpolate between regular sinusoidal and strongly nonlinear relaxation oscillations and demon-
strate that more pronounced nonlinearity induces multi-chimera states with multiple incoherent
domains. We show that the stability regimes for multi-chimera states and the mean phase velocity
profiles of the oscillators change significantly as the nonlinearity becomes stronger. Furthermore,
we reveal the influence of time delay on chimera patterns. V
C2015 AIP Publishing LLC.
[http://dx.doi.org/10.1063/1.4927829]
The investigation of coupled oscillatory systems is an im-
portant research field bridging between nonlinear dy-
namics, network science, and statistical physics, with a
variety of applications in physics, biology, and technol-
ogy.
1,2
The analysis and numerical simulation of large
networks with complex coupling schemes continue to
open up new unexpected dynamical scenarios. Chimera
states are an example for such intriguing phenomena;
they exhibit a hybrid structure combining coexisting
domains of both coherent (synchronized) and incoherent
(desynchronized) dynamics and were first reported for
the well-known model of phase oscillators.
3,4
In this pa-
per, we investigate the influence of the local dynamics of
the oscillators upon the resulting chimera patterns. Using
the Van der Pol oscillator, which is a model allowing for
a continuous transition between sinusoidal and strongly
nonlinear relaxation oscillations by tuning a single pa-
rameter, we show that multi-chimera patterns with mul-
tiple incoherent domains are promoted by increasing the
nonlinearity of the local oscillator dynamics.
I. INTRODUCTION
The last decade has seen an increasing interest in chi-
mera states in dynamical networks.
5–13
It was shown that
they are not limited to phase oscillators but can be found in a
large variety of different systems including time-discrete
maps,
14
time-continuous chaotic models,
15
neural sys-
tems,
16–18
and Boolean networks.
19
Moreover, chimera
states were found in systems with higher spatial dimen-
sions.
7,9,13,20–22
Together with the initially reported chimera
states, which consist of one coherent and one incoherent do-
main, new types of these peculiar states having multiple
incoherent regions,
16,18,23–25
as well as amplitude-medi-
ated,
26,27
and pure amplitude chimera and chimera death
states
28
were discovered.
In many systems, the form of the coupling defines the
possibility to obtain chimera states. The nonlocal coupling
has generally been assumed to be a necessary condition for
chimera states to evolve in coupled systems. However,
recent studies have shown that even global all-to-all cou-
pling
27,29–31
and more complex coupling topologies allow
for the existence of chimera states.
32–37
Furthermore, time-
varying network structures can give rise to alternating chi-
mera states.
38
The important question of the main features that give
rise to chimera states in coupled systems has been widely
discussed, but no conclusive answer has been given yet. In
systems of phase oscillators, the value of the phase lag pa-
rameter a, which occurs in the coupling function, is crucial.
In nonlocally coupled systems, the range of the coupling and
its strength play the key role. If the local dynamics of each
unit is described by a two- or higher dimensional system,
then the interaction scheme between the units plays an im-
portant role, i.e., which variable is coupled to which variable
of the other nodes. Chimera states have also been shown to
be robust against inhomogeneities of the local dynamics and
coupling topology.
37,39
Possible applications of chimera states in natural and
technological systems include the phenomenon of unihemi-
spheric sleep,
40
bump states in neural systems,
41,42
power
grids,
43
or social systems.
44
Many works considering chi-
mera states have mostly been based on numerical results. A
deeper bifurcation analysis
45
and even a possibility to control
chimera states
46,47
were obtained only recently.
The experimental verification of chimera states was first
demonstrated in optical
48
and chemical
49,50
systems. Further
a)
b)
Electronic mail: anna.zakharova@tu-berlin.de
c)
d)
Electronic mail: j.siebert@mailbox.tu-berlin.de
e)
1054-1500/2015/25(8)/083104/8/$30.00 V
C2015 AIP Publishing LLC25, 083104-1
CHAOS 25, 083104 (2015)
experiments involved mechanical,
51
electronic
52,53
and elec-
trochemical
54,55
oscillator systems, Boolean networks,
19
the
optical comb generated by a passively mode-locked quantum
dot laser,
56
and superconducting quantum interference
devices.
57
In previous investigations of chimera states, usually the
character of the local node dynamics has been considered as
fixed. In the current study, we address the issue of the impact
of the local dynamics. We analyze the properties of chimera
states, when the dynamics of individual oscillators smoothly
changes from sinusoidal to nonlinear relaxation oscillations.
For this reason, we choose the Van der Pol oscillator to
describe the dynamics of each node. The Van der Pol oscilla-
tor
58
has a long history of being used in both the physical
and biological sciences, as a generic model for electrical cir-
cuits
59
and action potentials of neurons, respectively.
II. THE MODEL
In our study, we consider a system of nonlocally
coupled Van der Pol oscillators with ring topology, where
each element of the system interacts with a fixed range of its
neighbors in both directions
€
uk¼e1u2
k
_
ukuk
þr
2RX
kþR
j¼kR
b1ujuk
ðÞ
þb2_
uj_
uk
ðÞ
;(1)
with k¼1; :::; Nwhere all indices are taken modulo N,eis
the bifurcation parameter of the individual oscillator, r
denotes the strength of the coupling, Ris the number of
coupled neighbors (in each direction), and b
1
,b
2
are the
interaction parameters. For such a form of coupling, it is con-
venient to consider the ratio r¼R=N, which we denote as a
coupling range. The uncoupled Van der Pol oscillator has a
stable trivial steady state u¼0 for e<0 and exhibits a
supercritical Hopf bifurcation at e¼0. Here, we consider
e>0.
Introducing a new variable vk¼_
uk, Eq. (1) can be
rewritten in the form of a two-dimensional system
_
uk¼vk
_
vk¼e1u2
k
vkuk
þr
2RX
kþR
j¼kR
b1ujuk
ðÞ
þb2vjvk
ðÞ
:(2)
The form of the coupling in the system Eq. (1) or (2) is
inspired from biological systems, describing interaction of
the cells or pattern generation in locomotion.
60,61
A similar
form of the coupling is also used in mechanics.
62
The cross-
couplings between the u- and the v-variable play an impor-
tant role, and they were shown to be necessary for the
existence of chimera and multi-chimera states in systems of
nonlocally coupled FitzHugh-Nagumo oscillators.
16
The dynamics of the system Eq. (2) is determined by
five parameters: edefines the dynamics of each individual
unit, and the parameters r,R,b
1
, and b
2
specify the coupling.
In order to find suitable values for some of the system param-
eters in the regime where Eq. (2) can describe chimera states,
we will use the experience from simpler systems of coupled
Kuramoto phase oscillators. For this reason, we transform
our system using the phase averaging technique on a rotating
frame for slowly varying amplitude r
k
and phase h
k
:ukðtÞ¼
rkðtÞsinðtþhkðtÞÞ and vkðtÞ¼rkðtÞcosðtþhkðtÞÞ.Asa
result, we obtain the approximate system
_
rk¼e
8rk42r
eR2Rþ1
ðÞ
b2
r2
k
þr
4Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b2
1þb2
2
qX
kþR
j¼kR
rjcos hkhjþa
_
hk¼r
4R2Rþ1
ðÞ
b1
r
4Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b2
1þb2
2
qX
kþR
j¼kR
rj
rk
sin hkhjþa
(3)
with a¼arctanðb1=b2Þ;b2>0 and k¼1; :::; N:
The parameter ain the system (3) can be associated
with the phase lag parameter in the systems of coupled phase
oscillators.
4
This parameter is crucial for the appearance of
chimera states in the phase oscillator network. In Ref. 8,it
was shown that a value of the phase lag parameter close to
but slightly less than p=2 allows for the existence of chimera
states.
In the following, using the experience from the phase
oscillator network, we fix the interaction parameters to be
b1¼1 and b2¼0:1, such that a1:47 is close to p=2.
With this parameter choice, we will focus further on the orig-
inal system Eq. (2) and vary the parameter ethat defines the
type of local dynamics of each element, as well as the cou-
pling parameters rand rdescribing the strength and the
range of the coupling, respectively.
III. THE IMPACT OF LOCAL DYNAMICS
Varying the bifurcation parameter eresults in a change
of the character of the local node dynamics. If eis small, the
uncoupled individual elements of the system perform har-
monic oscillations on a limit cycle, which is approximately a
circle. With increasing e, the individual limit cycle becomes
distorted and changes its form to relaxation oscillations.
Figure 1demonstrates examples of chimera states for
the system of N¼1000 elements, e¼0:2, and decreasing
coupling range. The upper panels depict snapshots of the var-
iables u
k
for fixed time T¼50 000. As initial conditions, we
use randomly distributed phases on the circle u2þv2¼4,
i.e., around the limit cycle of the uncoupled system, which is
approximately a circle of radius 2. One can clearly distin-
guish coherent and incoherent domains, a characteristic sig-
nature of chimera states. Elements that belong to the
incoherent domain are scattered along the limit cycle, as
shown with red points in the bottom panels of Fig. 1, where
the black line denotes the limit cycle of the uncoupled unit
with corresponding value e¼0:2. The individual nodes per-
form a nonuniform rotational motion, but neighboring oscil-
lators are not phase-locked. To illustrate this, the middle
083104-2 Omelchenko et al. Chaos 25, 083104 (2015)
panels of Fig. 1show the mean phase velocities for each os-
cillator calculated as xk¼2pMk=DT;k¼1; :::; N, where
M
k
is the number of complete rotations around the origin
performed by the k-th node during the time interval DT.
Throughout the paper, we use DT¼50 000 for the calcula-
tion of the mean phase velocities x
k
, corresponding to sev-
eral thousand periods. The values of x
k
lie on a continuous
curve and the interval of constant x
k
corresponds to the
coherent domain, where neighboring elements are phase-
locked. This mean phase velocity profile is a clear indication
of chimera states and similar to the case of coupled
Kuramoto phase oscillators.
3,4
In addition to chimera states with one incoherent domain
[Fig. 1(a)], we observe chimera states with multiple incoher-
ent domains shown in Figs. 1(b)–1(d), i.e., multi-chimera
states. In the following, we will use the notation n-chimera
for a chimera state with ncoherent and nincoherent
domains. In analogy with networks of phase oscillators, here
we observe chimera states where the number of incoherent
domains is even. The number of incoherent domains
increases with decreasing coupling range.
Figure 2(a) shows the stability regimes for chimera
states with one and multiple incoherent domains in the plane
of coupling range rand coupling strength rfor e¼0:2.
Indeed, for large coupling range, we observe the stability re-
gime for chimera states with one incoherent domain, and
regimes for chimera states with two, four, and six incoherent
domains follow subsequently with decreasing coupling
range. The overlaps of these regimes are characterized by
multistability, when each of the chimera states can be
obtained in the system depending on the choice of the initial
conditions. The regimes shown in the diagram are obtained
by starting from the chimera states shown in Fig. 1, and
using this pattern as initial condition for the neighboring pa-
rameter set, and so forth with a step size of Dr¼0:01 and
Dr¼0:01. Black squares denoted by A-D show values of
the parameters ðr;rÞthat correspond to the examples pre-
sented in Figs. 1(a)–1(d), respectively.
For larger coupling strength r, we observe coherent
states in the system (2). They are characterized by the wave-
number Kdefining the number of maxima (minima) in the
spatial profile, and K¼0 corresponds to complete in-phase
synchronization. The wavenumber increases with decreasing
coupling range, and exemplary snapshots are shown in the
insets of Fig. 2(a). For large coupling strength r, the system
is characterized by high multistability, and depending upon
initial conditions, one can obtain coherent solutions with dif-
ferent wavenumbers. In our system, there exist two different
FIG. 1. Snapshots of the variables u
k
(upper panels), mean phase velocities x
k
(middle panels), and snapshots in the phase space (u
k
,v
k
) (bottom panels, limit
cycle of the uncoupled unit shown black). (a) r¼0.35, r¼0:05, (b) r¼0.2, r¼0:09, (c) r¼0.13, r¼0:09, (d) r¼0.1, r¼0:09. Other parameters:
N¼1000, b1¼1;b2¼0:1, and e¼0:2.
FIG. 2. Stability regimes for multiple
chimera states. (a) e¼0:2, black squares
marked by A-D show parameter values
corresponding to panels (a)–(d) in Fig. 1.
The insets show snapshots of coherent
spatial profiles for parameter values A
(r¼0.35, r¼0.12; K¼0), B
(r¼0.2,
r¼0.28; K¼1), C
(r¼0.13, r¼0.34;
K¼2), D
(r¼0.1, r¼0.38; K¼3); (b)
e¼0:4. Other parameters as in Fig. 1.
083104-3 Omelchenko et al. Chaos 25, 083104 (2015)
types of chimeras, amplitude-mediated chimeras and pure
phase chimeras, and these are generated by different bifurca-
tion mechanisms. The amplitude-mediated 2-, 4-, 6-chimeras
(Fig. 2(a), snapshots in Figs. 1(b)–1(d), top panel) are gener-
ated from smooth, completely coherent spatial profiles of
wavenumbers K¼1, 2, 3, respectively, by a coherence-
incoherence bifurcation with decreasing coupling strength as
indicated by the insets in Fig. 2(a). At the onset of chimeras,
the smooth coherent profiles break up into spatially coherent
domains corresponding to the upper and the lower parts of
these profiles and incoherent domains in between. Therefore,
these incoherent domains occur in pairs (2-, 4-, 6-chimeras).
Such coherence-incoherence bifurcations have also been
observed for other local dynamics, e.g., logistic maps and
R€
ossler systems,
14
the cosine map,
48
and Stuart-Landau
oscillators.
28
In contrast, the pure phase chimeras (1-chimera
in Figs. 2(a) and 2(b)) arise from completely in-phase
synchronized (K¼0) profiles. The shift of the regimes for
(multiple) n-chimeras to smaller coupling range rwith
increasing nis typical for various nonlocally coupled
systems.
14–16,28,48
For small e, the limit cycle of each individual Van der
Pol oscillator is close to a circle, corresponding to sinusoidal
oscillations, and the similarities to the chimera states in a
system of phase oscillators are clearly revealed. However,
the hybrid solutions we observe in the system of coupled
Van der Pol oscillators demonstrate chimera behavior both
for phases and amplitudes. This can be seen in the bottom
panels of Fig. 1, where red dots denoting the snapshot of all
nodes deviate in their amplitudes slightly from the limit
cycle of the uncoupled unit (black line).
Figure 2(b) depicts the stability regimes for chimera
states in the system (2) with e¼0:4. Compared to the case
of e¼0:2, the regimes for the chimera states with multiple
incoherent domains become larger, and chimera states in this
case can be obtained for a wider range of coupling strength
r. The reason why the regions in Fig. 2(b) (e¼0:4) are
larger than those in Fig. 2(a) (e¼0:2) is related to the fol-
lowing qualitative argument: If the coefficient eof the non-
linear term in Eq. (1) is increased, the coefficient rof the
coupling term has to be scaled up accordingly to balance the
nonlinear term. Hence, as eis increased, the chimera regions
extend to larger values of r.
When the parameter eis increased, the limit cycle of the
individual uncoupled Van der Pol oscillators deforms, and
the dynamics on the cycle becomes slow-fast type. Further
increasing eleads to strongly nonlinear relaxation oscilla-
tions. This will be discussed in the following.
Figure 3(a) depicts the stability regimes for system (2)
with e¼0:8. Compared to the cases of smaller e, there are
several qualitative differences in the stability regimes of chi-
mera states. First, chimera states can be observed for a much
larger range of coupling strength r. Second, chimera states
with one incoherent part cannot be observed in the system
any more: for large coupling range, we observe chimera
states with four incoherent parts, and furthermore with
decreasing coupling range, the multiplicity of the incoherent
domains of the chimera states increases. Black triangles
denoted by E, F, and G show values for the parameter pairs
ðr;rÞthat correspond to examples of chimera states depicted
in Figs. 4(a)–4(c), respectively.
The peculiarity of the diagram presented in Fig. 3(a) is
the presence of two separate regimes for chimera states with
four incoherent domains. Analyzing this diagram in more
detail, one can see that there are two qualitatively different
regions. The first region appears for large coupling strengths
and contains stability regimes for chimera states with two
and four incoherent domains (yellow region containing point
E and blue region). These states can be characterized by
strong amplitude dynamics, and the maximum values of the
mean phase velocity profile correspond to the coherent
domains of chimera states. The example presented in Fig.
4(a) [corresponding to point E in Fig. 3(a)] depicts these fea-
tures. Compared with Fig. 1for small e, we notice that the
chimera states shown there also show distinct variations
along the limit cycle, and the coherent domains of the chi-
mera states correspond to the maximum in the mean phase
velocity profiles.
The second, qualitatively different part of the diagram in
Fig. 3(a), includes three regions for small coupling strengths
(yellow including point F, green, and gray). These regions
form a similar sequence with increasing multiplicity of the chi-
mera starting from four incoherent parts. However, they ex-
hibit a qualitative difference. Inspecting Figs. 4(b) and 4(c),
which show examples that correspond to the parameter pairs
ðr;rÞdenoted by F and G, one can notice that the amplitude
dynamics becomes weaker in these cases, and the network so-
lution is close to the limit cycle of the uncoupled node shown
as black line in the bottom panels. Moreover, the minimum of
the mean phase velocities profiles now corresponds to the
coherent domains of the chimera states.
The difference between Fig. 4(a) on one hand and Figs.
4(b) and 4(c) on the other hand is due to two different types
of chimeras. The 2-, 4-, 6-chimeras in Figs. 2(a),2(b) and
Fig. 3(a) (point E), and the chimeras in Figs. 3(F,G,H,I,J)
belong to two different types of chimeras: Figs. 2(a),2(b)
FIG. 3. Stability regimes for multiple
chimera states. (a) e¼0:8, black trian-
gles marked by E-G show parameter
values corresponding to the panels (a)-
(c) in Fig. 4; (b) e¼1:5, black circles
marked by H-J denote parameter val-
ues corresponding to the panels (a)-(c)
in Fig. 5. Other parameters as in Fig. 1.
083104-4 Omelchenko et al. Chaos 25, 083104 (2015)
and Fig. 3(a), point E (corresponding to phase portraits
shown in Figs. 1(b)–1(d) and Fig. 4(a)) correspond to
amplitude-mediated chimeras with strong amplitude-phase
coupling, whereas Fig. 3(a), points F, G and Fig. 3(b) (corre-
sponding to phase portraits shown in Figs. 4(b),4(c) and
Figs. 5(a)–5(c)) correspond to pure phase chimeras similar to
the ones found for Kuramoto phase oscillators, and since the
phase oscillator model can generally be obtained from
amplitude-phase models in the weak coupling limit, they
occur in the stability diagram (Fig. 3(a)) only for small cou-
pling strength (points F,G), as opposed to the amplitude-
mediated chimeras (point E). This difference is visible in the
phase portraits of Figs. 1,4,5(bottom panels), where the
spread of the various oscillators around the cycle of the
uncoupled oscillator (black cycle) is large for amplitude-
mediated chimeras and very small for pure phase chimeras
where the phase of the cycle is the only dynamical degree of
freedom. The difference also shows up in the smaller ampli-
tude variation of the mean phase velocity in the middle pan-
els of Figs. 4(b) and 4(c) (pure phase chimeras) as compared
to Fig. 4(a) and in the inverted x
k
profiles: the coherent
regions correspond to the minima (Figs. 4(b) and 4(c)) and
maxima (Fig. 4(a)), respectively.
Further increase of the parameter eof individual Van
der Pol oscillators leads to an even stronger deformation of
the limit cycle. In the ðr;rÞparameter plane, the stability
regimes for chimera states with four and more incoherent
domains can be observed as shown in Fig. 3(b) for e¼1:5.
The effect of the coexistence of two qualitatively different
types of chimera states is not present there any more, in con-
trast to the case of e¼0:8. Only the second type of the chi-
mera states is observed in the systems now, and the stability
regimes are enlarged towards larger coupling strengths.
Figure 5depicts examples of multi-chimera states that
correspond to the parameter pairs ðr;rÞdenoted by H, I, and
J in Fig. 3(b). The coherent domains of the chimera states
correspond to the minimum of the mean phase velocity pro-
file, and all oscillators stay very close to the limit cycle of
the single uncoupled unit, thus the amplitude dynamics of
the chimera states in the systems with large eis not as pro-
nounced as in the networks with small e.
We conclude that the nonlinearity of the local dynamics
indeed strongly influences chimera states in system (2). The
character of the amplitude dynamics, the frequencies of the
oscillators belonging to the coherent and incoherent domains
of the chimera states, i.e., the mean phase velocity profiles,
and the stability regimes in the coupling parameter plane
undergo a qualitative change with variation of the parameter
e. Stronger nonlinearity (larger e) results in the dominance of
multi-chimera states with weak amplitude dynamics.
IV. TIME-DELAYED COUPLING
Together with the character of the local dynamics, the
coupling between the individual units plays an important
role for the properties of the chimera states. Time-delayed
coupling if compared to the instantaneous one represents a
more realistic way to model the interaction between the
FIG. 4. Snapshots of the variables u
k
(upper panels), mean phase velocities x
k
(middle panels), and snapshots in the phase space (u
k
,v
k
) (bottom panels, limit
cycle of the uncoupled unit shown black). (a) r¼0.17, r¼0:8, (b) r¼0.35, r¼0:3, (c) r¼0.25, r¼0:2. Other parameters as in Fig. 3(a).
083104-5 Omelchenko et al. Chaos 25, 083104 (2015)
coupled units. Usually, the coupling range and strength influ-
ence the multiplicity of coherent domains in chimera states.
However, it has been shown for phase oscillator networks
that time delay can also induce multi-chimeras.
23
The exis-
tence of chimera states in systems with time-delayed cou-
plings has been also reported in Refs. 50,63–66.In
particular, for coupled phase oscillator systems, it has been
found that chimeras are robust to small time delays and delay
distributions
64
and can become unstable depending on the
value of delay.
65
Here, we consider a model that includes not
only phase but also amplitude dynamics and show how time
delay in the coupling affects chimera states that exist in the
undelayed system. We demonstrate that by varying the delay
value, one can both conserve and eliminate chimera patterns.
Let us consider Eq. (2) modified by time-delayed coupling
_
ukðtÞ¼vkðtÞ
_
vkðtÞ¼e½1u2
kðtÞvkðtÞukðtÞ
þr
2RX
kþR
j¼kR
fb1½ujðtsÞukðtÞ
þb2½vjðtsÞvkðtÞg (4)
wth k¼1; :::; Nmodulo N, where sis the delay time.
FIG. 5. Snapshots of the variables u
k
(upper panels), mean phase velocities x
k
(middle panels), and snapshots in the phase space (u
k
,v
k
) (bottom panels, limit
cycle of the uncoupled unit shown black). (a) r¼0.4, r¼0:1, (b) r¼0.22, r¼0:1, (c) r¼0.17, r¼0:1. Other parameters as in Fig. 3(b).
FIG. 6. Space-time plots of u
k
for different values of time delay: (a) s¼1, (b) s¼3, (c) s¼6. Other parameters: N¼1000, b1¼1;b2¼0:1, r¼0.4,
r¼0:1;e¼1:5. Initial conditions as shown in Fig. 5(a). Transients of 2000 time units are skipped.
083104-6 Omelchenko et al. Chaos 25, 083104 (2015)
Using the chimera state with four incoherent domains,
shown in Fig. 5(a), as initial condition, i.e., as the history in
the interval ½s;0, we fix all system parameters correspond-
ing to this solution and show exemplary space-time patterns
of Eq. (4) for different time delays. The period of a single
uncoupled oscillator is close to 2p, and we neglect the transi-
ents of 2000 time units. For small time delay (s¼1), we
observe a coherent traveling wave solution shown in Fig.
6(a). When the time delay is close to half the oscillation pe-
riod (s¼3), the chimera pattern is stable and we continue to
observe a chimera state with four incoherent domains as
shown in Fig. 6(b). A larger time delay (s¼6), which is
close to the period of a single oscillator, leads to complete
synchronization of all oscillators, see Fig. 6(c). Figure 7
depicts snapshots of the variable u
k
(upper panels), the same
snapshots in the phase plane (u
k
,v
k
) together with the limit
cycle of uncoupled oscillator, and the corresponding mean
phase velocity profiles (middle panels), for the solutions
shown in Fig. 6. The explanation for the effect of delay is
that the delay time interacts with the intrinsic timescale (os-
cillation period) giving rise to resonance phenomena as
found generally for delayed feedback control of steady
states, deterministic limit cycles, and noise-induced oscilla-
tions, if the delay is an integer multiple or a half-integer mul-
tiple of the intrinsic timescale.
67
Delay has a favorable effect
on chimeras if sis a half-integer multiple, and a favorable
(stabilizing) effect on the synchronized oscillations if it is an
integer multiple, and may induce traveling waves if it fits
with neither condition. In case of the chimera (Fig. 7(b)), the
delay leads to much longer transients, so that with the same
length of time interval used for the calculation of the mean
phase velocity as without delay, the profiles are more
smeared out, but qualitatively similar.
Our numerical evidence shows similar results for other
values of the bifurcation parameter e. For chimera states with
one or two incoherent domains and sinusoidal character of
the oscillations, small delay can lead not only to traveling
wave solutions but also to chimera states with higher number
of incoherent domains.
These examples demonstrate that time delay introduced
in the coupling can either suppress or preserve the chimera
patterns depending upon the value of the delay time relative
to the intrinsic oscillation period.
V. CONCLUSION
In the current study, we have demonstrated how the
character of the local oscillator dynamics influences chimera
states in networks of nonlocally coupled Van der Pol oscilla-
tors. Changing the bifurcation parameter of the single oscil-
lators allows us to interpolate continuously between
sinusoidal and strongly nonlinear relaxation oscillations. We
have shown that nonlinearity facilitates multi-chimera states.
For small values of the bifurcation parameter e(sinusoi-
dal oscillations), chimera states are characterized by lower
multiplicity and more pronounced amplitude dynamics, and
the maxima in the mean phase velocity profiles correspond
to the coherent domains. Moving towards the relaxation
FIG. 7. Snapshots of the variables u
k
(upper panels), mean phase velocities x
k
(middle panels), and snapshots in the phase space (u
k
,v
k
) (bottom panels, limit
cycle of the uncoupled unit shown black). (a) s¼1, (b) s¼3, (c) s¼6. Other parameters: N¼1000, b1¼1;b2¼0:1, r¼0.4, r¼0:1, e¼1:5.
083104-7 Omelchenko et al. Chaos 25, 083104 (2015)
oscillation regime, with increasing e, leads to a higher multi-
plicity of chimera states, but weaker amplitude dynamics. In
contrast to the previous case, the coherent domains corre-
spond to the minima of the mean phase velocity profiles, i.e.,
the profiles are flipped. We have also found that time delay
in the coupling strongly affects the chimera patterns in the
system and can lead to chimera suppression and the forma-
tion of traveling waves and complete synchronization.
We have presented (multi-) chimera states of different
types: (i) pure phase chimeras, which are similar to those
found for Kuramoto phase oscillators or weakly coupled
amplitude-phase models and (ii) amplitude-mediated chime-
ras with strong amplitude-phase coupling.
Our findings give new insight into the intriguing phe-
nomena of chimera states and demonstrate that the character
of the local dynamics has a strong influence on the chimera
patterns in the whole network. These results could be useful
from the point of view of applications dealing with different
kinds of oscillators, as they can be realized, e.g., in electronic
circuits.
ACKNOWLEDGMENTS
This work was supported by Deutsche
Forschungsgemeinschaft in the framework of Collaborative
Research Center SFB 910. P.H. acknowledges support by
BMBF (grant no. 01Q1001B) in the framework of BCCN
Berlin.
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