Chimera-like states in two distinct groups of identical populations of coupled Stuart-Landau oscillators

4.3 MB·

by  K. Premalatha

Chimera-like states in two distinct groups of identical populations of coupled
Stuart-Landau oscillators
K. Premalatha1, V. K. Chandrasekar2, M. Senthilvelan1, M. Lakshmanan1
1Centre for Nonlinear Dynamics, School of Physics,
Bharathidasan University, Tiruchirappalli - 620 024, Tamil Nadu, India.
2Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering,
SASTRA University, Thanjavur -613 401,Tamilnadu, India.
We show the existence of chimera-like states in two distinct groups of identical populations of
7 globally coupled Stuart-Landau oscillators. The existence of chimera-like states occurs only for a
1 small range of frequency difference between the two populations and these states disappear for an
0 increaseofmismatchbetweenthefrequencies. Herethechimera-likestatesarecharacterizedbythe
2 synchronized oscillations in one population and desynchronized oscillations in another population.
Wealso find that such states observed in two distinct groups of identical populations of nonlocally
n
coupled oscillators are different from the above case in which coexisting domains of synchronized
a
and desynchronized oscillations are observed in one population and the second population exhibits
J
synchronized oscillations for spatially prepared initial conditions. Perturbation from such spatially
4
prepared initial condition leads totheexistence of imperfectly synchronized states. An imperfectly
2
synchronizedstaterepresentstheexistenceofsolitaryoscillatorswhichescapefromthesynchronized

group in population-I and synchronized oscillations in population-II. Also the existence of chimera
]
O state is independent of the increase of frequency mismatch between the populations. We also find
thecoexistenceofdifferentdynamicalstateswithrespecttodifferentinitialconditionswhichcauses
A
multistability intheglobally coupledsystem. Inthecaseof nonlocal coupling,thesystemdoes not
. show multistability except in thecluster state region.
n
i
l PACSnumbers: 05.45.Xt,89.75.-k
n
[

1 I. INTRODUCTION otherchimerapatternslikechimeradeathstates[18,39–
v 41] and phase-flip chimera states [42] have been recently
2 explored. Forinstance,inRef. [4],Abramsetal. studied
2 Dynamical systems in nature are rarely isolated. In-
thestabilityofthechimerastatesintwointeractingpop-
9 teractions between the dynamical units often give rise
ulations of globally coupled phase oscillators where one
6 to new phenomena which can be exploited in differ-
population is synchronized and the other population is
0
ent contexts in physical, biological and social sciences
. desynchronized. It does not contain synchronized oscil-
1 [1]. Among them, the recently discovered emergent phe-
lators from both the populations. They have also shown
0 nomenon of chimera states has been in the center of
that the chimera states are stable and stationary, and
7
recent research towards the study of coupled networks.
1 that these states become periodically breathing chimera
In the chimera state the synchronous and asynchronous
: statesduetostabilitychange. LaterPikovskyandRosen-
v behaviors are observed simultaneously in a network of
blum [29] have observed quasiperiodic chimera states in
i coupled identical oscillators. Chimera state was first re-
X the same model considered in Ref. [4] by nonuniformly
ported by Kuramoto et al. in nonlocally coupled phase
r distributing the initial conditions. On the other hand,
a oscillators [2, 3]. The fascinating phenomenon was ini- globally clustered chimera (GCC) states were reported
tially found in nonlocal coupling configurations [4–15].
bySheeba,ChandrasekarandLakshmanan[9]inthecase
Howevermorerecentresultsrevealthatthesystemswith
ofdelaycoupledphaseoscillators. Theyhaveshownthat
globally [16–19] and locally coupled oscillators [20] are
theexistenceofsuchstatescanbeachievedbytuningthe
alsocapableofshowingsuchphenomenon. Recentlysuch
delay parameter.
states have also been explored in experiments with op-
Whereas many of the studies related to the chimera
tical [21], electronic [22], electrochemical [23], and me-
patternshavebeenfocusingonthenetworksofphaseos-
chanical oscillators [24].
cillators, in this work we address the question as to how
Sucharemarkablephenomenonhasbeensubsequently the emergenceofchimerasariseintwodistinct groupsof
studied for a variety of systems including chemical oscil- identical populations of Stuart-Landau oscillators where
lators[25],neuronmodels[26],planaroscillators[27],het- variations in amplitudes also occur. Here the interac-
erogeneous systems [28], oscillators with more than one tion between the oscillators are equally shared by both
populations [29–35] as well as in hierarchical networks the populations. One important fact to be noted here is
[36] and also in a variety of system structures such as thatthefrequenciesoftheoscillatorsareidenticalwithin
rings,networkswithtwoandthreeoscillatorpopulations their population but have a finite difference among the
[37, 38] and two dimensional map lattices [8]. Several twogroups. Oursystemisdifferentfromthesystemcon-
2
sidered in Ref. [4] where the authors assumed that the the populations in the above globally coupled oscillators
oscillatorswithin a populationarecoupledstronglywith with two parameter phase diagrams. In Sec. IV we dis-
eachotherthanthatwiththeneighboringpopulationand cuss how the dynamics of nonlocally coupled oscillators
also differ from the system considered in Ref.[28], where get affected by the presence of frequency mismatch be-
Laing assumed the two populations of phase oscillators tween them. Finally we critically summarize our results
with nonidentical natural frequencies. Recently, in ref. in Sec. V.
[43], the authors have identified transition from cluster
states to extensive chaos and then to incoherent state
while decreasing the system parameter in an identical II. MODEL OF TWO DISTINCT COMPETING
populationofgloballycoupledStuart-Landauoscillators. IDENTICAL POPULATIONS OF
STUART-LANDAU OSCILLATORS
Our motivation in this paper is to investigate the oc-
currence of chimera-like states by considering two dis-
tinct groups (each one is characterized by its own fre- We consider two distinct populations of globally cou-
quency) of identical populations of Stuart-Landau oscil- pled Stuart-Landau oscillators for our analysis, by as-
lators which are coupled through (i) global and (ii) non- suming the same common frequencies within a popula-
local couplings. We study the characteristic behavior of tion and different frequencies between the populations.
the oscillators by fixing the frequency of the oscillators The governing equations are specified by the following
in one population while varying the frequency of the os- set of equations,
cillatorsin the otherpopulation. We find the occurrence
ofdifferentdynamicalstatesincludinglocalsynchroniza- z˙j(1,2) =(1+iω(1,2))zj(1,2)−(1−ic)|zj(1,2)|2zj(1,2)
tion, localcluster states, chimera-like states, globalclus- (1,2)
+ε(z−z ), (1)
ter states and globalsynchronizationin globally coupled j
oscillators. Among these dynamical states, chimera-like
where z(1,2) =x(1,2)+iy(1,2), z =(1/N) N z . There
states occur only for a sufficient range of frequency mis- j j j Pk=1 k
areM oscillatorsin the population-I andthey arenum-
match and all the oscillators are coupled through equal 1
beredasz(1) whileM oscillatorsareinthepopulation-II
coupling interaction. In contrast to the chimera states j 2
found in a single array of coupled identical oscillators, and they are numbered as z(2). M +M =N and N is
j 1 2
the present case of chimera-like states (where synchro- thetotalnumberofoscillators. Inourstudy,weconsider
nized oscillations occur in one population while another boththepopulationsto beofequalsize, M =M =M.
1 2
populationisinafullydesynchronizedstate)existintwo For a discussion on M 6= M , see the last paragraph of
1 2
distinct groupsof identical populations with a frequency Sec. IIIbelow. InEq. (1),εisthecouplingconstantand
mismatch between them. It occurs neither for a very cisthenonisochronicityparameter,ω(1) andω(2) arethe
low mismatch nor for a larger mismatch of frequencies. natural frequencies of the oscillators in the two popula-
These states are different from chimera states found in tions of the network. This model can also be considered
coupledidentical oscillatorsin whichtotal number ofos- as a single population with two clustered networks. In
cillatorsaresplitintocoexistingdomainsofsynchronized oursimulations,wechoosethenumberofoscillatorsN to
and desynchronized oscillations [18]. beequalto200andinordertosolvetheequation(1),we
Further,wealsofindthatthechimerastatesexhibiting usethefourthorderRunge-Kuttamethodwithtimestep
synchronized and desynchronized oscillations in nonlo- 0.01 and the initial state of the oscillators(x(1,2), y(1,2))
j j
callycoupledtwodistinctgroupsofidenticalpopulations are distributed with uniform random values between -1
aredifferentfromthegloballycoupledcaseforthereason and +1. We have also verified that the results are in-
that in the previous case, synchronized and desynchro- dependent of increasing the number of oscillators in the
nized oscillators exist in one population while another population.
population shows synchronizedoscillations. Existence of
such a state is independent of frequency mismatch be-
tween the two populations under nonlocal coupling and III. GLOBAL INTERACTION
exist for spatially prepared initial conditions. Perturba-
tionfromsuchinitialstatesleadstoimperfectlysynchro- To start with, by assuming ω 6= 0, ω 6= 0 in (1), we
1 2
nized states in this region. In globally coupled system, consider a small frequency difference between the two
wefindthemultistabilityinthesystemdependingonthe populations such that ω(2) − ω(1) = ∆ω = 0.5. To
initial conditions used. On the other hand, a nonlocally explore the results of different dynamical behaviors, we
coupled system does not show any multistability region present the associated space-time plots in Figs. 1(a-e).
except in the cluster region. For small values of the coupling strength of the oscilla-
The structure of the paper is organized as follows: In tors, both the populations are individually synchronized
Sec. II we describe our model of two distinct groups of withtwodifferentaveragefrequencies(frequencyprofiles
identical populations of globally coupled oscillators. In ofthe oscillatorsarediscussedbelowinFig. 3),see Figs.
Sec. III we discuss the existence of various dynamical 1(a) and 1(f). This means that the oscillators are hav-
states by introducing the frequency mismatch between ing the same average frequency within their population
3
x(1,2) x(1,2) x(1,2) x(1,2) x(1,2)
10 j 1 10 j 1 10 j 1 10 j 1 10 j 1
(a) (b) (c) (d) (e)
t 0 t 0 t 0 t 0 t 0
0 -1 0 -1 0 -1 0 -1 0 -1
1 j 200 1 j 200 1 j 200 1 j 200 1 j 200
1 1 0 1
(f) (g) 1.0 (h) (i) (j)
2) 2) 2) 2) 2)
1, 1, 1, 1, 1,
(xj (xj (xj (xj (xj
-1
-1 0 -0.7 -1
1 100 200 1 100 200 1 100 200 1 100 200 1 100 200
j j j j j
FIG. 1: (Color online) Space-time plots of the variables x(1,2) depicting (a) local synchronization for ε = 0.1, (b) local
j
cluster synchronization for ε=0.8, (c) chimera-like state for ε=1.1, (d) global cluster synchronization for ε=1.8, (e) global
synchronizationforε=2.8. Inallthesecaseswehavechosen∆ω=0.5andc=3.0. Further,(f)-(j)representthecorresponding
snapshots of the variables xj for Figs. (a)-(e). Note that in the above, x(j1,2), j=1,2,...,100 correspond to population-I, that is
x(1), and x(1,2), j=101,102,...,200 correspond to population-II, that is x(2). One can obtain similar plots for y(1,2) also, which
j j j j
are not presented here.
strength of coupling interaction to ε = 1.1 one observes
1.0 1.0
(a) (b) that the mismatch between the frequencies of two pop-
ulations causes asynchronous oscillations among the os-
5
0 5 cillators in population-I while the population-II remains
1 y
y
synchronizedwhichdenotesthe existenceofchimera-like
-1.0 -1.0 states. In contrast to the chimera states found in a sin-
glearrayofcoupledidenticaloscillators,thepresentcase
-1.0 x 1.0 -1.0 x 1.0
105 5
of chimera-like states exist in two identical populations
1.0 1.0 with a frequency mismatch between them. Figure 1(c)
(c) (d)
represents the space-time plot for the chimera-like state.
5 1 ItssnapshotinFig. 1(h)confirmsthecoexistenceofspa-
5 5
y y tiallycoherentandincoherentdistributionsofoscillators.
In this state the high frequency population remains syn-
-1.0 -1.0
chronized having the same average frequency while the
-1.0 x 1.0 -1.0 x 1.0
55 51 low frequency population is desychronizedwith its oscil-
latorshavingdifferentaveragefrequencies. Onincreasing
εtoε=1.8wenotethatthesystementersintotheglobal
cluster state as shown in Fig. 1(d). In this state the
FIG. 2: (Color online) Phase portraits of the representa- total number of oscillators in both the populations are
tive oscillators in chimera-like states: (a) For synchronized
grouped into two groups (Fig. 1(i)). These two groups
oscillator (z ) with segment Λ (red/grey dots), (b) for
105 105 are entrained to a common average frequency but with
desynchronized oscillator (x ) with data set A (red/grey
5 5 different amplitudes. We can observe the globally syn-
dots), (c) for desynchronized oscillator (x ) with segment
55
chronized oscillations (with same frequency, amplitude
Λ (red/grey dots), (b) for desynchronized oscillator (z )
55 51
with data set A (red/grey dots). and phase) among the oscillators for ε beyond ε=2.2.
51
On order to confirm the existence of chimera-like
states, we analyze the phase-locking behaviour of the
but they are different for different populations. By in- synchronizedand desynchronizedoscillatorsthroughthe
creasing the coupling strength to ε = 0.8 the system of following localized set approach [44]. To illustrate this
oscillators in population-I splits into two groups. These approach, we choose one of the representative oscilla-
two groups are oscillating synchronously with same fre- tors from both the synchronized and desynchronized
quency though there is a finite phase difference between groups. First, we construct a data set A by observ-
5
the two groups, while the population-II remains oscil- ingthe desynchronizedoscillatorz wheneverthe trajec-
5
lating synchronously. They are clearly illustrated with tory of synchronized oscillator z crosses the segment
105
the space-time plot in Fig. 1(b) and with a snapshot Λ (= x >0.8,y ≈0.0). In Fig. 2(a), the at-
105 105 105
of the variables in Fig. 1(g). Hence these states are tractor of synchronized oscillator z is shown by black
105
designated as local cluster states. Upon increasing the curveandred/greylinerepresentsthesegmentΛ . The
105
4
attractor of the desynchronized oscillator z5 is depicted (A)
5.5
by black curve and the data set A5 is shown by the (a)II III IV V VI 4.0(b)
red/grey dots in Fig. 2(b) and in which the spread-
ing of the data set over the trajectory clearly illustrates (2)fj
2)
that the deysnchronized oscillator is not in phase with 1, 3.6
the synchronized oscillator. Further, we also illustrate (fj 100 15j0 200
3.0(c)
the non-phaselockingbehaviourbetween the desynchro-
I
nized oscillators in Figs. 2(c) and (d). The attractor 2.5 (1)fj
and corresponding segment of one of the randomly cho- 0.0 2.25 4.5
2.7
sen representative oscillators z are shown by the black ε 1 50 100
55 (B) j
curve and red/grey line in Fig. 2(c). In Fig. 2(d),
1.0 (i)-a 1.0 (i)-b 1.0 (ii)-a 1.0 (ii)-b
we can observe that the data set Λ corresponding to
51
the oscillator z is spread over the trajectory z . This (1)yj (2)yj (1)yj (2)yj
51 51
also clearly illustrates that there is no phase locking be- -1.0 -1.0 -1.0 -1.0
tween the desynchronized oscillators. Thus we confirm -1.0 xj(1) 1.0 -1.0 xj(2) 1.0 -1.0 xj(1) 1.0 -1.0 xj(2) 1.0
that the existence of chimera-like states which is illus- 1.0 (iii)-a 1.0 (iii)-b 1.0 (iv)-a 1.0 (iv)-b
trated in Figs. 1(c) and (h). The chimera-like states are (1)yj (2)yj (1)yj (2)yj
distinguished from the intermittent chimera states [46] -1.0 -1.0 -1.0 -1.0
and quasi-periodic chimera states [3] that are observed -1.0 xj(1) 1.0 -1.0 xj(2) 1.0 -1.0 xj(1) 1.0 -1.0 xj(2) 1.0
intwopopulationsofcoupledrotatorswherethe authors 1.0 (v)-a 1.0 (v)-b 1.0 (vi)-a 1.0 (vi)-b
have considered broken-symmetryconditions realized by (1)yj (2)yj (1)yj (2)yj
initializing thefirstpopulationwithidenticalphasesand
-1.0 -1.0 -1.0 -1.0
frequencieswhiletheyarerandomforsecondpopulation. -1.0 xj(1) 1.0 -1.0 xj(2) 1.0 -1.0 xj(1) 1.0 -1.0 xj(2) 1.0
Notethattheimportantfacttobenotedhereisthatthe
frequencies of the oscillators are identical within their
populationbuthaveafinitedifferenceamongthegroups. FIG. 3: (Color online) A(a) Average frequencies of the oscil-
Also the initial conditions are chosen uniformly between latorsasafunctionofthecouplingstrengthfortheparameter
−1 and +1. values ∆ω = 0.5, c = 3.0. Enlarged structures for a specific
value of ε = 2.05: (b) Average frequencies of the synchro-
To give more details about the nature of the dynami- nizedoscillators(population-II).(c)Frequenciesofthedesyn-
cal states (discussed above), we present the frequency of chronized oscillators (population-I).B. Corresponding to the
the oscillators as a function of the coupling strength (ε) evolution in the complex plane, snapshots of the variables
in Figs. 3 (A). We calculate the average frequencies of (x(j1,2),yj(1,2)): (i)-a and b correspond to desynchronization
for ε = 0.01 (Region-I), (ii)-a, b correspond to local syn-
the oscillators by using the expression f = 2πΩ /∆T,
j j
chronization for ε=0.1 (Region-II), (iii)-a, b represent local
where Ω is the number of maxima in the time series x
j j cluster states for ε=0.8 (Region-III),(iv)-a, b represent the
over a time interval ∆T. Also for a clear understanding
chimera-likestatesforε=1.1(Region-IV),(v)-a,brepresent
about the dynamical regions, we have plotted the evo-
the global cluster states for ε=1.8 (Region-V) (vi)-a, b cor-
lution of the dynamical variables in the complex plane responds to global synchronization fro ε = 2.8 (Region-VI).
(z(1,2) = x(1,2) +iy(1,2)) and their respective snapshots Trajectories areshown byblacksolid lines andred/grey dots
j j j
ofthevariables(x(1,2),y(1,2))areprojectedontothecom- represent thesnapshotsof thevariables zj(1,2) in thecomplex
j j plane.
plex plane in Fig. 3 B(i-vi). In Fig. 3 A(a), initially for
the range between 0.0 < ε < 0.05, the oscillators are
having same frequency within populations but they are
oscillating incoherently as shown in Figs. 3B(i)-a and quency while the incoherentoscillatorsare havingdiffer-
(i)-b. For the range 0.05<ε<0.8, we can observe local ent frequencies which are clearly shown in Figs. 3 A(b)
synchronizations, that is each population is individually and A(c). Figs. 3 B(iv)-a and (iv)-b also validate the
synchronized with two different frequencies represented chimera-like behaviour where the incoherent oscillators
by region-II in Fig. 3 A(a). Figs. 3 B(ii)-a and (ii)-b are randomly distributed in the complex plane while the
clearly show the individual synchronization. In the re- distribution of the synchronizedoscillators are shownby
gion0.8<ε<1.4,wecanobservelocalclusterstates. In the red/grey dot. Further for the values of ε between
thisstate,oscillatorsinthelowfrequencypopulationsep- 2.25 to 2.8, one can find global cluster states where the
arate into different synchronized groups (Figs. 3 B(iii)-a frequenciesofalltheoscillators(inboththepopulations-
and(iii)-b). Buttheirfrequenciesarethesamewhilethe IandII)arethe sameasillustratedinregion-VofFig. 3
high frequency population remains synchronized with a A(a)(andfurtherconfirmedbyFigs. 3B(v)-aand(v)-b).
singlefrequencyasshownintheregion-IIIinFig. 3A(a). Finally, beyond the value ε = 2.8, both the populations
Upon increasing ε to 1.4 < ε < 2.25, we can observe aregloballysynchronizedtoacommonfrequency,seere-
chimera-likestates in the region-IVin Fig. 3 A(a) where gion VI in Fig. 3A(a) and Figs. 3 B(vi)-a and (vi)-b.
the coherentoscillatorsareoscillatingwith the same fre- By considering the various dynamical states described
5
CS
(c)
1
2)
1,
(R, r
0
GC
LC 1000 1150
time
5
(b) (A) (d)
1 DS LC CS GC 1
(1,2)R, r c 2.5 III II3 ε’c (1,2)R, r
1
0 LS 0
GS
1000 time 1150 ε II2 1000 time 1150
c
0
0 1.0 2.0
LS ε GS
5
(a) (B) (e)
1 1
LC GC
1,2) DS I ε’c 1,2)
(R, r c 2.5 II II3 (R, r
1
0 LS GS 0
1000 time 1150 ε 1000 time 1150
c
0
0 2.0 4.0
ε
FIG.4: (Coloronline)Thetwoparameterphasediagraminthe(ε,c)parametricspace: (A)for∆ω=0.5and(B)for∆ω=1.5.
Boundaryεc isobtainedfromthecriticalvalueofthecouplingstrengthgiveninEq. (5). Thecurveε′c isobtainedfromtheEq.
(6) and the other boundaries are obtained numerically. Region LS represents the local synchronization, region GS represents
theglobalsynchronization,region LCindicatesthelocal clusterstates, region CSrepresentsthechimera-likestatesandregion
GCrepresentstheglobalclusterstates. Region-Iisthemultistabilityregionbetweenthelocalsynchronizationandlocalcluster
states. Regions II is the multistability region between complete synchronization and local cluster states. Region II is the
1 2
multistability region between complete synchronization and chimera states. Region II is the multistability region between
3
completesynchronization andglobalclusterstates. Figs. (a)-(e)presenttheglobal synchronyorderparameterR(blue/dark-
grey solid curve) and local order parameter r(1) (red dashed curve) and r(2) (black solid curve) for the local synchronization,
local cluster states, chimera-like states, global cluster states and global synchronization, respectively.
above, except for the chimera-like state and global clus- θ˙ (1,2) =ω(1,2)+cr2(1,2)
j j
ter state, all the oscillators in each of the populations
N (1,2)
possess the same amplitude (note that in the local clus- + ε rk sin(θ(1,2)−θ(1,2)) (2)
terregionalsothe amplitudesr = x2+y2 remainthe N Xr(1,2) k j
j q j j k=1 j
same). Hence in order to study analytically the regions
If we substitute r(1,2) = r(1,2) = constant so that
of local and globalsynchronizations,we assume that the k j
amplitudes of all the oscillators remain the same to find r˙ (1,2) =0, we will get the phase equation as
j
the regionsoflocalandglobalsynchronizations. By sub-
stituting zj(1,2) =rj(1,2)eiθj(1,2) in equation (1) we will get θ˙ (1,2) =ω(1,2)+c(1−ε)+ εc N cos(θ(1,2)−θ(1,2))
j N X j k
k=1
r˙(1,2) =r(1,2)−r3(1,2) ε N
j j j + sin(θ(1,2)−θ(1,2)).(3)
N X j k
N
ε k=1
+ (r(1,2)cos(θ(1,2)−θ(1,2))−r(1,2))
N X k k j j Complex order parameter within each population is de-
k=1
6
sRcerpiblaecdinbgythze(1s,2u)m=mart(i1o,2n)etieψr(m1,2b)y=th(e1c/oMm)pPlexMk=o1rdeeiθrk(1p,2a)-. tahlseoraergiisoenssaLqSu,eLstCio,nCwS,hGetChearntdheGcSh,imreesrpae-clitkiveesltya.tTeshaerree
rameter, equation (3) becomes, robust to increasing or decreasing the mismatch of fre-
quencies. To address this question, we now present the
θ˙ (1,2) =ω(1,2)+c(1−ε)+ εcr(1,2)cos(θ(1,2)−ψ(1,2)) existence of dynamical states in the parametric space
j N j (ε,∆ω).
ε
+ r(1,2)sin(θ(1,2)−ψ(1,2)).(4)
N j
2.0 2.0
(a) (b)
In the case of local synchronization region (LS), oscil- DS LC GC DS LC GC
lators within each group are identical and so the lo-
ε
cal order parameter r(1,2) = 1 (red dashed and black ω 1.0 LS c ω 1.0 LS I
solid line) while the global order parameter R < 1 ∆ I ∆ εc
(Nbolutee/tdhaartkt-hgeregylosboalildolridneer)pwahriacmheitsesrhioswdnefiinneFditgh.ro4u(ag)h. εc’ GS ε’c CS GS
Z = ReiΨ = 1 2 r(σ)eiψ(σ), where R measures the 0 0
degree of globa2lPsyσn=ch1rony for the entire system. In the 0 1.0 2 0 2.0 4
ε ε
regionGS,wecanobservetheglobalsynchronizationand
thevaluesofbothRandr(1,2) areunityasshowninFig.
4(e). The critical value of coupling strength at which
global synchronization occurs is given by [32] FIG. 5: (Color online) Phase diagrams for globally coupled
system (1) by varying the values of ∆ω and ε (a) for c=1.5
εc =∆ω/(2cosα), (5) avanldue(bo)ftfohrecco=up3l.i0n.gBstoruenndgathrygεivceinsionbEtaqi.ne(d5)f.roTmhetchuercvreitεi′caisl
c
where α = arctan[c]. From the linear stability analy- obtainedfrom Eq. (6)andtheotherboundariesareobtained
sis of Eq. (4), we will get the eigenvalues for in-phase numerically. The region DS represents the desynchronized
state, LS is the local synchronization region, GS shows the
synchronized states [32] as
globally synchronized state, GC shows the state correspond-
ingtoglobalclusterregion,CSshowschimerastates,andLC
λ± =−εcosα−εcos(±∆ψ+α)<0, (6)
shows thelocal cluster state. The region-I is the multistabil-
ityregionbetweenthelocalsynchronizationandlocalcluster
where ∆ψ = arcsin[∆ω/(2εcosα)]. The choice λ = 0
+
leads to the curve ε′ shown by the red/grey curve in states.
c
Fig. 4(A) and it matches with the numerically obtained
boundary. Different dynamical regions in Fig. 4(A) for
∆ω = 0.5 and (B) for ∆ω = 1.5 are identified with the 11111 1
help of strength of incoherence [45] for every choice of ε (((((aaaaa))))) (b)
andcvalues. Somedetailsonstrengthofincoherenceare
givenin Appendix A. Fig. 4(a) represents localsynchro- nnnnn n
nizationwherethelocalorderparametersr(1) =1(black
solid curve), r(2) = 1 (red dashed curve) and global or-
der parameter R < 1 (blue/dark-grey solid curve). In 00000.....11111 0.1
the regions corresponding to local cluster states (LC), 11111 ccccc 1111100000 0.01 0–ε.1 1
chimera-like states (CS) and global cluster states (GC),
wecanobservethelocalorderparametersr(2) =1(black
solid curve), r(1) < 1 (red dashed curve) and global or-
der parameter R<1 (blue/dark-grey solid curve) which FIG. 6: Log-log plots: (a) The value of n (= M1) as a func-
are depicted with Figs. 4(b), (c) and (d). In globally tion of the nonisochronicity parameter c forMfi2xed ε values
synchronized region, both r(1,2) and R take the value (ε = 0.7 for local synchronization, ε = 0.9 for local clus-
ter states, ε = 1.5 for chimera-like states, and ε = 2.3 for
unity which is shown in Fig. 4(e). The region-I is the
globalclusterstates)and(b)thevalueofn(= M1)asafunc-
multistability region between local synchronization and tion of the coupling strength ε¯for a fixed c=3M.02. Note that
local cluster states. The region-II1 is the multistability ε¯=ε−εe,whereεeistheearliestcouplingstrength. Dots(•)
regionbetweenthelocalclusterstatesandcompletesyn- representthenumericaldataandthecorrespondingbestfitis
chronization. Alsotherearisesamultistabilityregionbe- represented by the (black) curve for the chimera-like states.
tween chimera-like states and complete synchronization Similarly the local synchronization, local cluster states and
in region-II . Similarly, region-II represents the mul- global cluster states are represented by the solid line with
2 3
tistability region between the global cluster states and (N), ((cid:7)) and ((cid:4)), respectively. Note that εe = 0.05 for lo-
complete synchronization. The regions LS, LC, CS, GC cal synchronization, εe = 0.8 for local cluster state, εe = 1.4
and GS are always stable, that is in these regions even for chimera-like state and εe = 2.25 for global cluster state.
Otherparameter value: ∆ω=0.5
fora smallperturbationofthe initial stateofthe oscilla-
tors from the completely synchronized solution leads to
7
(1,2) (1,2) (1,2) (1,2)
x x x x
10 j 1 10 j 1 10 j 1 10 j 1
(a) (b) (c) (d)
0 0 0 0
t t t t
0 -1 0 -1 0 -1 0 -1
1 j 200 1 j 200 1 j 200 1 j 200
1 1 1 1
(e) (f) (g) (h)
2) 2) 2) 2)
1, 1, 1, 1,
(xj (xj (xj (xj
-1 -1 -1 -1
1 j 200 1 j 200 1 j 200 1 j 200
4.2 3.85 3.85 4.1
(i) (j) (k) (l)
2) 2) 2) 2)
1, 1, 1, 1,
(fj (fj (fj (fj
3.7 3.83 3.84 4.0
1 j 200 1 j 200 1 j 200 1 j 200
FIG.7: (Coloronline)Space-timeplotsofthevariablesx(1,2): (a)localsynchronizationforε=0.2,(b)imperfectlysynchronized
j
state (solitary state) for ε=0.35, (c) cluster state for ε=0.4, (d) global synchronization for ε=1.1 with c=3.0,r =0.4 and
∆ω=0.5. Figs. (e)-(h)representthesnapshotsofthevariablex(1,2) forFigs. (a)-(d). Figs. (i)-(l)representthecorresponding
j
frequency profiles of the oscillators for Figs. (a)-(d). Again here oscillators j=1,2,...,100 correspond to group-I (x(1)) while
j
oscillators numbered j=101,102,...,200 represent group-II (x(2)).
j
Inordertoanalyzetheglobalpictureofthedynamical of chimera states. Thus we conclude that for only a suf-
system under the influence of frequency mismatch, we ficientlylargevalueoffrequencymismatch(∆ω)onecan
plotted the two parameter phase diagrams in the para- induce the onset of chimera-like states. We can also ob-
metric space (ε,∆ω) for two different nonisochronicity serve that the chimera-like states inherit from out of lo-
parameter values, that is c =1.5 and c =3.0, which are cal cluster states. Also the chimera-like states mediate
illustrated in Figs. 5(a) and (b), respectively. In these between the local cluster states and global cluster states
figures, the curve ε is obtained from the critical value andarequitedifferentfromtheresultsinref. [48],where
c
of the coupling strength for globally synchronizedregion theauthorshavedemonstratedtheemergenceofchimera
given in Eq. (5). The curve ε′ is obtained from Eq. states in a system of globally coupled Stuart-Landauos-
c
(6) and the other boundaries are obtained numerically. cillators. Such chimera states have inherited properties
These regions are identified with the help of strength of from the cluster states in which they originate. In Figs.
incoherence [45]. In Fig. 5(a), we can observe that for a 5(a)and(b),thelocallysynchronizedregion(LS)andlo-
range of frequency mismatch 0 < ∆ω < 0.5, the system calcluster(LC)regionsarealwaysstable. Theregion-Iis
ofoscillatorsattainsglobalsynchronizationthroughlocal the multistability region between the local synchroniza-
synchronization. Further increasing the frequency mis- tionandlocalclusterstates. Inthisregion,forinitialcon-
match beyond ∆ω > 0.5, the system attains global syn- ditions near the globally synchronized state one is lead
chronization, through local synchronization, local clus- to complete synchronization among the oscillators. Also
ter and global cluster states. From Fig. 5(b), we can one can check numerically that the chimera-like states
find that for a sufficiently small value of frequency mis- and globally clustered states are always stable.
match (∆ω ≤ 0.34), global synchronized state is medi-
The above discussed results are observed for the case
ated through local cluster states as well as through the
whereboththepopulationsareofequalsize(M =M ).
globalcluster statesfromindividualsynchronizedstates. 1 2
There arise a question about the robustness of the dy-
Here,onecannotobservetheexistenceofchimerastates.
namical states for the case when M 6= M . To illus-
Interestingly, we can observe the onset of chimera states 1 2
trate this, we decrease the number of oscillators (M )
only in the range 0.35 < ∆ω < 1.57. Beyond this range 1
in population-I so that the number of oscillators (M )
of frequency mismatch one cannot observe the presence 2
in population-II becomes M = N −M . Then we find
2 1
8
the critical value of n ((that is n = M1)) at which there M = M), N is the total number of oscillators in both
occurs no qualitative change in the Mdy2namical states as the2 populations (M + M = N), r = P is the cou-
1 2 N
a function of nonisochronicity parameter c. Note that if pling range. When 1 < P < N/2, one has nonlocal
thevalueofnisunityitrepresentsthatboththepopula- coupling. In the present study, we are interested to an-
tions have equal number of oscillators. Fig. 6 (a) shows alyze the chimera states in two distinct groups of iden-
thelog-logplotforthecriticalvalueofnagainstthenon- tical populations of nonlocally coupled oscillators. Here
isochronicityparametercforfixedεvalue(wehavefixed theintra-populationsofoscillatorsaregloballyconnected
ε=0.7forlocalsynchronization,ε=0.9forlocalcluster and the oscillators are nonlocally connected with inter-
states, ε = 1.5 for chimera-like states, and ε = 2.3 for populations. For simplicity, we choose the coupling in-
global cluster states). We can observe that the critical teractionε = ε = ε inEq. (7) throughoutthis section.
1 2
value of n follows the power law relation n = pcq with In order to analyze the dynamics of the system under
c for the local synchronizationregionand the best curve nonlocal coupling, we present the spatiotemporal evo-
fitisobtainedforthevaluesp=0.23108andq =0.47019 lution, snapshots and the corresponding frequency pro-
which is shown by the solid line with (N) in Fig. 6 (a). filesf(1,2) ofthe oscillatorsforvariousvaluesofcoupling
Similarly the dynamical regions corresponding to local j
strength ε with fixed coupling range r = 0.4, ∆ω = 0.5
cluster, chimera-like states and global cluster states are
and nonisochronicity parameter c = 3.0 in Fig. 7. For
illustratedbythesolidlineswith((cid:7)),(•)and((cid:4)),respec-
a small value of coupling strength ε = 0.2, we can ob-
tively. Thevaluesofnfollowthepowerlawrelationwith
servethelocalsynchronization(Figs. 7(a)and(e))where
c for the parameter values p = 0.684708, q = 0.140745
the oscillators in each of the populations are entrained
in the local cluster region, p=0.0836, q =0.8498 in the
to two different common frequencies as shown in Fig.
chimera-like states region and p = 0.3381, q = 0.3717 in
7(i). Note that there is a finite frequency difference
the global cluster region. Similarly, Fig. 6 (b) is plotted
between them. Further increasing coupling strength to
for the criticalvalue ofn againstthe coupling strength ε¯
ε=0.35,inFigs. 7(b)and(f)wecanobservethesolitary
for a fixed c=3.0. Note that ε¯=ε−ε , where ε is the
e e statesinpopulation-Iwheresuchstatesrepresentthefact
earliest coupling strength which is different for different
thatsomeoftheoscillatorsescapefromthesynchronized
dynamicalregionsforafixednonisochronicityparameter.
group and exhibit non-phase coherent oscillations while
To obtainthe best curve fit, we subtract the consecutive
we find synchronization in population-II where the os-
εvalueswiththeε value(thatisε =0.05forlocalsyn-
e e cillators exhibit periodic oscillations. Figure 8(a) shows
chronization, ε =0.8 for local cluster state, ε =1.4 for
e e the phase portrait of the desynchronized oscillator (z )
3
chimera-likestate andε =2.25for globalclusterstate).
e which shows the non-phase coherent oscillations of this
Herealsowecanobservethatnfollowsthepowerlawre-
oscillator and in Fig. 8(b) we can observe the periodic
lation with coupling strength ε for the parameter values
oscillation of the synchronized oscillator z . Another
102
p = 0.916003, q = 0.411106 in the local synchronization
point is that the average frequency of the synchronized
region, p = 0.952379, q = 0.060111 in the local cluster
oscillatorsfrom both the populations are same while the
region, p = 1.00183, q = 0.599414 in the chimera-like
average frequency of the solitary oscillators are differ-
statesregionandp=1.08815,q =0.433495intheglobal
ent as shown in Fig. 7(j). By increasing the coupling
cluster region.
strength to ε = 0.5, the system of oscillators gets split
intotwogroupsofsynchronizedoscillationswithperiodic
motion which is shown in Figs. 7(c) and (g). These two
IV. NONLOCAL INTERACTION
groups are oscillating with common average frequency
(as in Fig. 7(k)) but with different amplitudes. These
Intheprevioussection,wehaveanalyzedtheexistence states are designated as cluster states. Upon increasing
of chimera-like states which represent asynchronous os- thecouplingstrengthtoε=1.1,alltheoscillatorsareen-
cillations in one population and synchronous oscillations trainedtoacommonfrequencywiththesameamplitude
in the second population in globally coupled oscillators while they are oscillating periodically. Their spatiotem-
under the influence of frequency mismatch. To investi- poral plot, snapshots and frequency profile are shown in
gate how the dynamics of nonlocally coupled oscillators Figs. 7(d), (h) and (l), respectively.
is affected by the frequency difference between the two
In order to explore the results in more detail, we
populations, we consider the dynamical equations of the
present the global picture in terms of a two phase di-
following form:
agrams in the (ε, c) parametric space for two different
∆ω values with fixed coupling range r = 0.4. Vari-
z˙j(1,2) =(1+iω(1,2))zj(1,2)−(1−ic)|zj(1,2)|2zj(1,2) ous dynamical states observed in Fig. 9 are identified
M j+P by making use of the concept of strength of incoherence
ε ε
+ 1 (z(1,2)−z(1,2))+ 2 (z(1,2)−z(2,1)), (7) [45] (see appendix for more details). From Fig. 9(a)
M X k j 2P X k j
k=1 k=j−P (for ∆ω = 0.5), we can find that the oscillators are ini-
tially for lowrangeof ε (andfor allvales ofc) exhibiting
where P is the total number of nearest neighbors, M desynchronized oscillations. By increasing the ε value,
is the number of oscillators in each population (M = thesystemofoscillatorsattainslocalsynchronizationfor
1
9
0.8 1.0 system from this specific choice of initial state of the
(a) (b)
oscillators, the system enters into the imperfectly syn-
y3 y102 cfrhormontihzeedstsattaeteosb.seTrhviesdkiinndgloobfaclhcimoueprlainsgt.atIenitshdeicffaesreenotf
globallycoupledsystem,onecanobservedesynchronized
-0.8 -1.0 oscillationsin population-I and synchronizedoscillations
-1.0 x3 1.0 -1.0 x102 1.0 inpopulation-II,whereasinthepresentcaseofnonlocally
coupled system, we find the coexistence of coherent and
incoherentdomains in population-I and coherent oscilla-
FIG. 8: Phase portraits of the representative oscillators
tions in population-II. By increasing the strength of the
in imperfectly synchronized state: (a) Represents the non-
coupling interaction beyond the IS region, the system
phase coherent oscillation of the solitary oscillator (z ) in
3
of oscillators enters into the cluster states (CL). Further
population-I. (b) Represents the periodic oscillation of the
oscillator (z ) in population-II. strengthening of the interaction strength leads to global
102
synchronization. IntheCLregion,thedistributionofini-
tial states near the synchronized solution leads to com-
8 8
(a) (b) plete synchronization. The cluster states are observed
for the initial conditions away from the synchronizedso-
lution. Similarly for the case of ∆ω = 1.5 illustrated in
c4 IS c4 LS IS Fig. 9(b), the system of oscillators follows similar tran-
CL sition routes as that of the case ∆ω =0.5 except for the
fact that the dynamical regions LS and IS are widened
LS CL
DS GS DS GS in the ε space.
0 0
0 0.5 1 0 0.5 1 3 3
(a) (b)
ε ε DS CL DS
LS IS LS IS
ω 1.5 ω 1.5
FIG.9: (Coloronline)Twophasediagramsinthe(ε,c)para- ∆ GS ∆ CL GS
metric space (a) for ∆ω = 0.5 and (b) for ∆ω = 1.5 with
r = 0.4 for the two population nonlocally coupled Stuart-
Landauoscillators. DSrepresentsthedesynchronizedregion, 0 0
LS represents the local synchronization region, IS shows the 0 0.75 1.5 0 0.75 1.5
imperfectlysynchronizedstates,CLisforclusterstateregion ε ε
and GS represents theglobally synchronized region.
FIG. 10: (Color online) Two phase diagrams in the (ε,∆ω)
parametric space (a) for c = 1.5 and (b) for c = 3.0 with
arangeofεvaluesforagivennonisochronicityparameter
r = 0.4 for a system of two distinct identical groups of non-
value. Upon further increasing the coupling interaction,
locally coupled Stuart-Landau oscillators. DS represents the
we can observe the solitary states in population-I and desynchronized region, LS represents the local synchroniza-
synchronized oscillations in population-II. This region is tion, IS shows the imperfectly synchronized states, CL is for
marked as imperfectly synchronized state (IS) in Fig. cluster state region and GS represents the globally synchro-
9. Note that the IS state is different from the imperfect nized region.
chimera state reported in ref. [49] and this state is char-
acterized by a certain small number of solitary oscilla- Nextweanalyzewhethertheemergenceofimperfectly
torswhichescapesfromthesynchronizedpartofchimera synchronized states (or chimera states for spatially pre-
state (where solitary oscillator represents a single repul- pared initial conditions) are robust to an increase of fre-
sive oscillator splitting up from the fully synchronized quency mismatchbetweenthe twopopulations. Toillus-
group). Such escaped oscillators oscillate with different tratethe results,we plottedthe two parameterphasedi-
average frequencies. The IS state is also different from agramsin the (ε, ∆ω) parametric space in Figs. 10(a,b)
the imperfect chimera state reported in ref. [50] where for c = 1.5 and c = 3.0 values, respectively, with fixed
the chimeras are characterizedby the coexistence of two coupling range r = 0.4. In Fig. 10(a), initially the
coherentoscillatorsandoneincoherentoscillator(i.e. ro- system of oscillators is desynchronized for small val-
tatingwithanotherfrequency). Ifthechimerastateloses ues of coupling strength. By increasing ε in the range
its ‘perfection’ in the sense that phases of the synchro- 0<∆ω <0.5,the systemattains globalsynchronization
nized oscillators become not equal, they still rotate with (GS) through the locally synchronized states (LS) and
the same average frequency. In this region, we can also imperfectly synchronized (IS) states. For ∆ω ≥ 0.5, the
observe the existence of chimera states in population-I system attains global synchronization (GS) through the
while synchronized oscillations in population-II for the clusterstates(CL)inadditiontothelocallysynchronized
specific choice of initial conditions. If we perturb the states (LS) and imperfectly synchronized states (IS). By
10
increasing the value of c to c = 3.0, the system attains observedincoupledidenticalpopulationswherethetotal
globalsynchronization(GS)throughthelocallysynchro- number ofoscillatorsaresplit into coexisting domainsof
nized states (LS), imperfectly synchronized states (IS) synchronized and desynchronized oscillations. We have
and cluster states (CL) for all values of frequency mis- found the results are robust for certain range of unequal
match between the populations as shown in Fig. 10(b). size populations (both the populations contain unequal
In Figs. 10(a) and (b), the regions IS and CL are mul- number of oscillators). In addition, we have found the
tistable regions for the reason that even for small per- existence of multistable regions depending on the initial
turbations of initial states from the synchronized state state of the oscillators.
leads to the existence of imperfectly synchronized states Further we have extended our study to analyze the
and cluster states in the regionsIS and CL, respectively. dynamics of nonlocally coupled oscillatorswhere we find
AlsointheISregion,wecanobservetheonsetofchimera the existence of chimera state for spatially prepared ini-
statesforspecificchoiceofinitialconditions. We canob- tial conditions. In contrast to the global coupling, ex-
serve that the existence of chimera state is independent istence of chimera states is independent of the increase
of frequency mismatch between the two populations un- of frequency mismatch between the two populations. In
der nonlocal coupling, whereas in the case of global cou- contrasttothecaseofglobalcoupling,multistabilitydoes
pledsystemtheexistenceofchimerastateisobtainedfor notexistinnonlocallycoupledsystemexceptintheclus-
only a range of frequency mismatch. We have also ana- ter state region. Finally, it may be an interesting open
lyzedtheresultsforawiderangeofnonlocalcouplingby problem to extend this study to a network of more than
plotting the two parameter phase diagram in the (ε, r) two populations, which we are pursuing currently.
parametric space with two different ∆ω values that is
∆ω =0.5 and ∆ω =1.5 for fixed c =3.0. Figures 11(a)
and(b)clearlyillustratetheexistenceofimperfectlysyn- Acknowledgements
chronizedstates(orchimerastatesforspatiallyprepared
initial conditions) and other dynamical states for a wide The work of VKC is supported by the SERB-DST
range of nonlocal coupling independent of the frequency Fast Track scheme for young scientists under Grant
mismatch. No.YSS/2014/000175. The work of MS forms part of
a research project sponsored by DST, Government of
0.45 0.45
(a) (b) India. ML acknowledges the financial support under a
DS DS NASI Platinum Jubilee Senior Scientist Fellowship pro-
GS GS gram.
r 0.22 CL r 0.22 CL
IS IS
LS LS Appendix A: CHARACTERISTIC MEASURE FOR
0.02 0.02 STRENGTH OF INCOHERENCE
0 1.5 3.0 0 1.5 3.0
ε ε
To identify the nature of different dynamical states,
we look at the strength of incoherence of the system
FIG.11: (Coloronline)Twophasediagramsinthe(ε,r)para- a notion introduced recently by Gopal, Venkatesan and
metric space (a) for ∆ω =0.5 (b) for ∆ω =1.5 with c=3.0 two of the present authors[45], that will help us to de-
for thesystem (7). DS represents the desynchronized region, tect interesting collective dynamical states such as syn-
LS represents the local synchronization, IS shows the imper- chronized state, desynchronized state, and the chimera
fectly synchronized states, CL is for cluster state region and state. For this purpose we introduce a transformation
GS representsthe globally synchronized region. w(1,2) = x(1,2) −x(1,2) [45], where j = 1,2,3,...,N. We
j j j+1
dividetheoscillatorsintoK binsofequallengthl =N/K
and the local standard deviation σ (m)(1,2) is defined as
l
V. CONCLUSION ml
1
σ (m)(1,2) =h( |w(1,2)−w(1,2)|2)1/2i ,
l l X j t
Insummary,wehaveillustratedtheexistenceofdiffer- j=l(m−1)+1
ent dynamical states in globally and nonlocally coupled m=1,2,...K.(A1)
systemsoftwodistinctgroupsofidenticalpopulationsof
oscillators. We have analyzed the impact of frequency From this we can find the local standard deviation for
mismatch between the two populations. An interest- everyK binsofoscillatorsthathelpstofindthestrength
ing feature here is that only for certain ranges of fre- of incoherence [45] through
quency mismatch, one can observe the onset of chimera-
like states in globally coupled network and the oscilla- K s(1,2)
S(1,2) =1− Pm=1 m ,s(1,2) =Θ(δ−σ (m)(1,2)),
tors are coupled with equal strength coupling interac- K m l
tion. These states are different from the chimera states (A2)