Chimera states for a globally coupled sine circle map lattice: spatiotemporal intermittency and hyperchaos
CChimera states for a globally coupled sine circle map lattice:spatiotemporal intermittency and hyperchaos
Joydeep Singha a) and Neelima Gupte b) Department of Physics, Indian Institute of Technology Madras, Chennai, 600036, India (Dated: 10 July 2019)
We study the existence of chimera states, i.e. mixed states, in a globally coupled sine circle map lattice, with differentstrengths of inter-group and intra-group coupling. We find that at specific values of the parameters of the CML, acompletely random initial condition evolves to chimera states, having a phase synchronised and a phase desynchronisedgroup, where the space time variation of the phases of the maps in the desynchronised group shows structures similar tospatiotemporally intermittent regions. Using the complex order parameter we obtain a phase diagram that identifies theregion in the parameter space which supports chimera states of this type, as well as other types of phase configurationssuch as globally phase synchronised states, two phase clustered states and fully phase desynchronised states. Weestimate the volume of the basin of attraction of each kind of solution. The STI chimera region is studied in furtherdetail via numerical and analytic stability analysis, and the Lyapunov spectrum is calculated. This state is identified tobe hyperchaotic as the two largest Lyapunov exponents are found to be positive. The distributions of laminar and burstlengths in the incoherent region of the chimera show exponential behaviour. The average fraction of laminar/burst sitesis identified to be the important quantity which governs the dynamics of the chimera. After an initial transient, thesesettle to steady values which can be used to reproduce the phase diagram in the chimera regime.
The study of chimera states, i.e. mixed states where syn-chronised and desynchronised dynamics coexist, has beenat the forefront of studies in nonlinear dynamics involv-ing both theoretical and experimental systems. A varietyof classes of chimera states, i.e. states which contain co-existing domains of distinct kinds of spatiotemporal be-haviour can be seen. These include multi-headed chimerastates, travelling chimera states, amplitude chimera states,twisted chimera states etc, and have been seen in coupledoscillator models such as the Kuramoto model,coupledGinzburg-Landau oscillators and other systems. Here, weinvestigate chimera and other states in a coupled sine cir-cle map lattice which is a discrete version of coupled os-cillator systems. The CML consists of two populationsof globally coupled identical sine circle maps with dis-tinct values for the intergroup and intragroup coupling.We observe spatiotemporally intermittent chimeras, i.e.states which consist of a synchronised subgroup, and astate where coherent (phase synchronised) and incoher-ent (phase incoherent) domains co-exist, at low valuesof the nonlinearity map parameter. Such STI chimerashave been observed earlier in coupled oscillators modelssuch as Stuart-Landau oscillators, Ginzburg-Landau os-cillators, coupled optical resonators, chemical reactionsetc. We analyse the STI chimera seen in the CML sys-tem by plotting the phase diagram of the system using theglobal order parameter, and identifying the region whereSTI chimeras can be seen. The basin stability of the STIchimera state, as opposed to other states e.g. fully synchro-nised states, fully desynchronised states and two clusterstates, which can be seen in the phase diagram is estab-lished. The linear stability analysis of the chimera regionis carried out, using analytic and numerical methods. The a) Electronic mail: [email protected] b) Electronic mail: [email protected]
Lyapunov exponents obtained via this analysis establishthat the STI chimera is hyperchaotic. Further, the pair-wise order parameter is used to distinguish laminar andburst sites, and the time evolution of the laminar and burstsites show that the fraction of laminar and burst sites inthe system reaches a steady state. The phase diagram ob-tained from these stationary states matches the phase dia-gram obtained from the complex order parameter exactly.We also study the distributions of laminar and burst sitesin the system, and find that they fall off exponentially dueto the globally coupled nature of the system.
I. INTRODUCTION
The chimera phase pattern is a remarkable spatiotempo-ral property found in spatially extended dynamical systems.This phase pattern has been seen in systems of coupled phaseoscillators and was recently discovered in coupled maplattice models . In the context of dynamical systems, the‘chimera’ state is defined to be a state with the characteris-tic stable coexistence of a synchronous group of oscillatorstogether with a desynchronised group of oscillators. Simi-lar dynamical behaviour was found in early studies of uni-hemispheric sleep and the asynchronous eye closure ofsea mammals, birds and reptiles. In addition to the phase cou-pled oscillator systems mentioned above, this kind of spatio-temporal behaviour has also been seen to exist in other os-cillator systems. These include non-locally coupled com-plex Ginzburg-Landau oscillators , delay-coupled rings ofphase oscillators , bipartite oscillator populations , Stuart-Landau oscillators , networks of Kuramoto oscillators ,coupled chemical oscillators , and mechanical oscillatornetworks . The detailed analysis of oscillator systems withdifferent kinds of coupling has been reviewed recently byOmel’chenko . a r X i v : . [ n li n . C D ] J u l Here, we study the existence of chimera states in a coupledmap lattice which is a discrete analog of coupled phase os-cillator system where both space and time are considered tobe discrete. The chimera phase state as well as other othermixed states were reported in specific systems of coupledmap lattices in both theoretical models and experimen-tal systems . The CML, used here, is of the form used inRefs. and consists of two populations of globally coupledidentical sine circle maps where the strength of the couplingwithin each population and that between the maps belongingto distinct populations take different values. Oscillator mod-els with two species of identical dynamical units, leading tochimera states have been explored earlier in refs. forphase oscillators and in Ref. for Fitzhugh-Nagumo oscilla-tors. The existence of chimera states in globally coupled sys-tems has also been reported for systems of Stuart-Landau os-cillators and for the complex Ginzburg-Landau equation .We note that different types of chimera states with interest-ing spatio-temporal behaviours have been studied in variouscontexts. These include multiheaded chimera states , trav-elling chimera states , multi-chimera states , twistedchimera states , and amplitude chimera states . It was alsoshown earlier that the specific CML which we study here cansupport another kind of mixed state, namely the splay-chimerastate where the coexistence of a phase synchronised group ofmaps and a phase desynchronised group of maps consistingof splay phase configurations was reported . In this paper,we report the existence of yet another kind of chimera statefor this system, where the evolution of random initial con-ditions in certain regions of the parameter space results in anew class of chimera solutions where the space time variationof the desynchronised group shows spatiotemporally intermit-tent behaviour. In addition to the chimera states describedhere, this system supports various other kinds of phase con-figurations viz. globally synchronised states, two phase clus-tered states, fully phase desynchronised states, etc. We definea complex order parameter for the entire system as well as foreach group. We show that the transition between these phaseconfigurations upon the change of the parameters can be iden-tified from these order parameters which take unique valuesfor each of these states. We thus obtain the phase diagram ofthe coupled map lattice and identify the regimes which sup-port chimera states of this type, and regimes which supportother phase configurations. Subsequent analysis focusses onthe chimera region of the phase diagram and its neighbour-hood. We note that chimeras with co-existing coherent andincoherent regions with spatiotemporally intermittent struc-tures have also been seen in systems of coupled oscillatorswith global and local coupling .We carry out the stability analysis of each solution thusidentified with special focus on the analysis of the chimerastates having spatiotemporally intermittent structures. Wenote that the phase space is high dimensional, leading tothe existence of multiattractor solutions at identical parametervalues. We find the relative volume of the basin of attractionof all these solutions including the STI chimera and its mir-rored version by estimating the fraction of initial conditionwhich evolve to each state. The linear stability analysis of the STI chimera can be car-ried out analytically due to the low values of the nonlinear-ity parameter. The values of the Lyapunov spectrum obtainedanalytically in this regime, match the numerically obtainedvalues. Two of the Lyapunov exponents of the system turnout to be positive, implying that the temporal evolution of theSTI chimera is hyperchaotic. Thus, this is one of the veryfew hyperchaotic chimera solutions seen so far . The lami-nar (coherent) and burst (incoherent) sites are identified usinga pairwise version of the global order parameter. The distri-bution of the length of laminar and turbulent segments showsexponential behaviour with a higher probability of longer tur-bulent segments. Due to the global nature of the coupling, thespatiotemporal evolution of the STI chimera depends only onthe fraction of laminar and turbulent sites in each subgroup.The average fraction of laminar and turbulent sites in eachsubgroup saturates to steady state values after an initial tran-sient. These steady state values are used to recreate the phasediagram in this regime. This phase diagram matches exactlythe phase diagram obtained via the global and subgroup or-der parameters, confirming that the average fraction of lami-nar and turbulent sites in each subgroup is the crucial factorwhich governs the dynamics of our system. We discuss theimplications of our results in practical contexts.Our paper is organised in the following manner: SectionII discusses the coupled sine circle map lattice model understudy. In section III, we introduce the complex order param-eters and obtain a phase diagram using their calculated val-ues. We also discuss here the variety of phase configurationsthat can be found when the system is evolved using randominitial conditions. Section IV discusses the basin stability ofeach of attractors including the chimera states. In section Vwe discuss the behavior of the chimera consisting of a phasesynchronised group and desynchronised group with spatio-temporally intermittent regions and obtain the Lyapunov ex-ponents in section V A. A method of identifying and labellingthe laminar and burst sites is outlined in section V B and thedistribution of laminar and burst segments is discussed in sec-tion V C. The evolution of the fraction of laminar and turbu-lent sites is discussed in section VI and the phase diagramis obtained in terms of their steady-state values. Section VIIsummarises our conclusions. II. THE MODEL
Here, we study a lattice of coupled sine circle maps, wherethe maps are distributed into two groups, which are globallycoupled, but with two distinct values for the intragroup andintergroup coupling. The evolution equation for a single sinecircle map is given by, θ n + = θ n + Ω − K π sin ( πθ n ) mod 1 (1)where θ is the phase of the map, 0 < θ < n is the timestep. The parameter Ω denotes the frequency ratio in the ab-sence of nonlinearity and K determines the strength of nonlin-earity. A single sine circle map shows Arnold tongues organ-ised by frequency locking and quasi-periodic behaviours . Itshows universality in the mode locking structure prior to boththe period doubling route to chaos and quasi-periodic route to chaos depending on the value of Ω . The evolution equa-tion for the coupled sine circle map lattice considered here isgiven by, θ σ n + ( i ) = θ σ n ( i ) + Ω − K π sin ( πθ σ n ( i )) + ∑ σ (cid:48) = ε σσ (cid:48) N (cid:34) N ∑ j = ( θ σ (cid:48) n ( j ) + Ω − K π sin ( πθ σ (cid:48) n ( j ))) (cid:35) mod 1 (2) Group one Group two ✓ (1) ✓ (2) ✓ (3) ✓ (1) ✓ (2) ✓ (3) FIG. 1: (color online) The schematic of the system of Eq. 2 with 3 maps ineach group. The intergroup coupling is shown by solid lines and theintra-group coupling is denoted by dotted lines.
The equation above defines the evolution of the i th map inthe group σ , where σ takes values 1 ,
2, and N is the numberof maps in each of the groups. We also define the couplingparameters to be ε = ε = ε and ε = ε = ε with theconstraint ε + ε =
1. Therefore, our model consists of twogroups of identical sine circle maps where N is the number ofmaps in each group. Each map in a given group is coupledto all the maps in its own group by the parameter ε whereasit is coupled to the maps in the other group by the parameter ε . We note that the evolution equation is completely sym-metric under interchange of the group labels, σ = ,
2. Thusthe system in equation 2 is controlled by three independentparameters, K , Ω , ε . A schematic of the CML of Eq. 2 withthree lattice sites in each group is shown in figure 1.This CML is a discrete version of globally coupled oscilla-tor models with two populations, which have been motivatedby biological examples of chimera states, such as the unihemi-spherical sleep patterns of sea mammals . In the oscilla-tor context a model consisting of two groups of identical Ku-ramoto oscillators representing each hemisphere of brain wasproposed by Abrams et al. and showed chimera states. TheCML which we discuss has a similar coupling topology, andcouples identical sine circle maps, which represent discreteversions of phase oscillator systems.The system under consideration has many degrees of free-dom with maps that are coupled globally with two groupswhich differ in their intergroup and intragroup coupling.As a consequence of this, different initial conditions gener-ally evolve to distinct attractors with different spatiotempo-ral properties; e.g. an initial condition where an identicalphase is assigned to each site will always evolve to a glob-ally synchronised state. In it was shown that an initial con-dition, where all the phases of the maps in one group areidentical while the maps in the other group are set to randomphases between zero and one, evolves to chimera states, clus- tered chimera states, clustered states etc. at different regionin the parameter space.Another initial condition with a sys-tem wide splay phase configuration was shown to evolve to asplay phase state, and to splay chimera states depending on theparameters . Initial conditions such as these break the sym-metry between the groups. Here, we explore this CML using avery general initial condition where the phases of each of themaps in both of the groups are randomly distributed betweenzero and one.We report that at certain parameter values, the fully randominitial condition evolves to a chimera state which consists ofa spatially phase synchronised group and a spatially and tem-porally phase desynchronised group (figure 2). At particu-lar values of K , Ω , ε and N we find a chimera phase statewith a purely synchronised subgroup where all maps in groupone belong to a phase synchronised cluster (see figure 2(a))whereas at other parameters we observe chimera states, wherethe spatially phase synchronised subgroup has defects, as thephases of a small fraction of circle maps do not belong to thesynchronised cluster (figure 2.(d)). We also see in figure 2.(b)and (e) that the space time variation of the desynchronisedgroup in both type of chimera states shows spatiotemporallyintermittent structures, as synchronised islands in the shapeof cones can be observed within the desynchronised phases.Other states can be seen at other parameter values which arediscussed in the next section. III. PHASE DIAGRAM
We note that the system is controlled by the parameters K , Ω , ε . Apart from this set of parameters, the system dy-namics also depends on the size, 2 N of the system and theinitial condition. We fix the size of the system at N = R n , R n , R n and the average phase, Ψ n , Ψ n , Ψ n defined respectively for each of the groups at timestep n as, R n exp (cid:0) i π Ψ n (cid:1) = N N ∑ j = exp (cid:0) i πθ n ( j ) (cid:1) (3) R n exp (cid:0) i π Ψ n (cid:1) = N N ∑ j = exp (cid:0) i πθ n ( j ) (cid:1) (4) pha s e map index
0 100 200 300map index 0 10 20 30 40 50 t i m e pha s e R , R Ψ , Ψ time (a) (b) (c) pha s e map index
0 100 200 300map index 0 10 20 30 40 50 t i m e pha s e R , R Ψ , Ψ time (d) (e) (f) FIG. 2: (color online) (a) The snapshot of the chimera state with a purely synchronised group is shown. The parameters are K = − , Ω = . ε = . N =
150 (b) The space time plot of the chimera state without the defects in the synchronised group. (c) The temporal variation of R , R , Ψ , Ψ for thechimera states with complete phase synchronisation in group one. (d) A snapshot of the chimera state with defects in the synchronised group. The parametersare K = − , Ω = . , ε = . , N = R , R and average phases, Ψ , Ψ (defined in Eq. 3 and 4 forgroup one and two respectively) for the chimera states with defects in the synchronised group shown in (d). In both (c) and (f), R and Ψ are shown in blackwhereas R and Ψ are denoted in red. R n exp ( i π Ψ n ) = N ∑ σ = N ∑ j = exp ( i πθ σ n ( j )) (5)It is clear that R n , R n becomes one when the phases of themaps in the corresponding group are synchronised at time step n . In that case, the phases at which the groups synchroniseare given by Ψ n , Ψ n respectively. Similarly their values be-come approximately zero when the phases are uniformly dis-tributed between zero and one. Similar conclusions can bedrawn for R n , Ψ n if the whole system is phase synchronisedor desynchronised. If all the maps are fully phase synchro-nised at a time step, then Ψ n and Ψ n become equal at thattime step, while R n , R n , R n become one. These properties ofthese quantities enable us to look for the chimera states of thetypes shown in figure 2.(c) and (d), as we vary the parameters K , Ω , ε .It is clear that the minimum number of time steps requiredfor the system to settle into chimera states of interest here isa function of the system size. Figure 3.(a) shows the varia-tion of the order parameters R n , R n , R n with time for differ-ent system sizes, N = , , , , R n rises to values above 0.8 afterthree hundred thousand time steps, and slowly tends to oneapproximately after three million time steps, while the sub-group order parameter R n becomes zero. Such values of thegroup wise order parameters imply the existence of chimeraphase configurations. The space time variation of the phasesof the maps at intermediate time steps show that the CML isin mixed configurations which are different (see Fig. 4.(a),(b)) from the chimera states under consideration. Here we al-ways evolve the system for 3 × iterations or more, in allour subsequent numerical calculations.We obtain a phase diagram for the parameter value Ω = .
27 and vary the parameters K , ε in the range 10 − < K < < ε <
1. At each values of these parameters we usea fixed set of initial phase values which are randomly dis-tributed between zero and one. We calculate R n , R n , R n for10 time steps and calculate the average after the system ofEq.2 is iterated for three million time steps. Figure 5 show thevalues of R , R and R respectively with the variation of K , ε at Ω = . R ≈ , R ≈
0. These show the existence of the chimera states (Fig. 2.(c))in a region in K , ε space approximately given by 0 . < ε < , − < K < − surrounded by other phase configurationsaround it whose snapshots are shown in Fig. 6. A magnified R R R , R timeN = 20N = 60N = 100N = 150N = 200 (a) t r an s i en t t i m e size (N) (b) FIG. 3: (color online) (a) The variation of the order parameters, R , R n , R n with time for N = , , ,
150 and 200. (b) Average transient time is plotted fordifferent sizes. An overall trend of increasing transient time can be observed.
0 50 100 150 200 250 300map index 0 10 20 30 40 50 t i m e pha s e (a)
0 50 100 150 200 250 300map index 0 10 20 30 40 50 t i m e pha s e (b) FIG. 4: (color online) The space time plot of the system after (a) 20000time steps and (b) 500000 time steps for N = K = − , Ω = . , ε = . version of this phase diagram around this region is shown inFig. 7. Five distinct types of phase configurations can befound in the phase diagram of Fig. 7. These are chimerastates, two clustered states, globally synchronised states andfully desynchronised states. The details of these dynamicalstates are as follows,1. Case 1 and case 2 : Chimera states (Fig. 2) : We ob-tain a chimera state when either R or R is one andthe value of the other quantity is near zero. We getthis condition at several of the parameter values for Ω = .
27. In particular when − < log K < − . < ε < .
0, at some parameters we find, case 1 : R = R ≈ R (cid:46) R ≈ R , R also shows this behaviour. The variationof Ψ and Ψ with time shows that the variation of theaverage phases of the phase synchronised and desyn-chronised group are qualitatively different (see Figs. 2.cand f). The mirrored version of these chimera stateswhere maps of group two phase synchronises whilemaps in group one become phase desynchronised aredenoted as Case 1" and case 2".2. Case 3 : Fully desynchronised states (Figs. 6.(c),(f)) : These are found at those parameter values where R , R , R are approximately zero. At these parametervalues, all the maps in both the groups are temporallyand spatially phase desynchronised. The temporal vari-ation of Ψ , Ψ suggest that the average phase of boththe groups are approximately periodic. They are ob-served approximately for log K < − K < − ε < . Case 4 : Two clustered states (Fig. 6.(a)) : We findthat R = R = − < log K < . < ε < R , R while the phases atwhich they synchronise are not equal as indicated by Ψ , Ψ (see Fig. 6.(d)). Figure 6.(d) also suggests thateach of these phase clusters do not synchronise to a tem-porally fixed phase value as can be seen from the varia-tion of the average phases Ψ , Ψ (see Fig. 6.b).4. Case 5 : Globally synchronised states (Fig. 6.(b)) :These are characterised by the order parameter valueswhen all three quantities, R , R , R are approximatelyone. They can be seen mostly above K ≈ − for ε below 0 .
8. The temporal variation of the average phasesof each of the groups, Ψ , Ψ in Fig. 6.(e) suggests thatall the maps are spatially phase synchronised at all timesteps although the phase at which they synchronise isnot a temporal fixed point similar to the temporal varia-tion of the two clustered state.
0 0.2 0.4 0.6 0.8 1 ε -10-8-6-4-2 0 l og ( K ) R (a)
0 0.2 0.4 0.6 0.8 1 ε -10-8-6-4-2 0 l og ( K ) R (b)
0 0.2 0.4 0.6 0.8 1 ε -10-8-6-4-2 0 l og ( K ) R (c) FIG. 5: (color online) The order parameters (a) R , (b) R and (c) R areplotted for the values N = , Ω = .
27. The color code for the values ofthe order parameter is indicated in the vertical bar in each plot. At eachparameter value we use a random initial condition and iterate the systeminitially for 4 × time steps, after which the order parameters ( R , R , R ) are calculated and averaged over 10 time steps. The region where chimerastates are seen is identified by the order parameter values R ≈ R ≈ In this paper we are mainly interested in the region of theparameter space where the chimera states are seen and its tran-sition to other phase configurations which are shown in Figs.6. Figure 7 shows that the fully desynchronised states seen inthe region − . < log K < − . < ε < . ε = .
8. The global phase desyn-chronised state seen between − < log K < − . . < ε < K increases be-yond − .
5. Between the parameter values − < log K < − . < ε < K , ε is better un-derstood from the variation of the order parameters R , R atdifferent cross sections of the phase diagram in the Fig. 7.Figure 8.(a) shows the variation of R n and R n with values of ε lying in the range between 0 .
65 and one for K = − . Itcan be seen that both the subgroup order parameters R , R take values near zero when ε is less than 0.8. These valuesof the order parameters suggest that the system is in a fullyphase desynchronised phase configuration for this range of ε and K values. When the parameter ε > . R = R ≈ ε → R (cid:46) R ≈
0. This indicates thatsome of the circle maps from the group one have phases thatdo not belong to the synchronised cluster at these values of ε .As discussed earlier, this indicates the presence of a chimeraphase state with defects in the synchronised group. We takeanother cross section of this phase diagram at the parameter ε = .
93 in Fig. 8.(b) that shows the variation R and R withlog K as it increases from − −
2. We find that the sub-group order parameters take values R = R ≈ K ≈ − . K ≈ − .
3. When log K lies between − .
36 and − . K . We find R and R both be-come one when log K > − .
7. In the next section we discussthe properties of the chimera states shown in Fig. 2.
IV. BASIN STABILITY
In the previous section, we have identified all the distinctspatiotemporal states found in different parameter regions ofthe phase diagram. We note that due to the large dimensionalnature of the phase space, multiattractor solutions exist, anddifferent initial conditions can go to different attractors at thesame parameter values. The fraction of random initial condi-tions that go to a given attractor constitute a measure of thevolume of its basin of attraction and also indicate the proba-bility for a random initial condition to evolve to the attractor.Recently, Menck et.al. showed that the volume of the basinof attraction of an attractor can be interpreted as a measure ofits global stability. In this section we discuss the basin sta-bility of the states seen in the phase diagram of the previoussection. The discussion of the basin stability of the chimerastate is particularly interesting.Fig. 7 shows a large parameter region of the phase diagram pha s e map index 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 pha s e map index 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 pha s e map index (a) (b) (c) R , R Ψ , Ψ time 0 0.5 1 R , R Ψ , Ψ time 0 0.5 1 R , R Ψ , Ψ time (d) (e) (f) FIG. 6: (color online) Snapshots of the (a) two cluster phase configuration at K = − , ε = .
93 (b) fully synchronised state at K = − , ε = .
45 and (c)fully de-synchronised state at K = − , ε = .
75 are shown. The variations of R , R , Ψ , Ψ are shown for (d) two phase clustered state, (e) globallysynchronised state and (f) fully desynchronised state at same parameters. Other parameters viz. Ω , N were kept fixed at 0 .
27 and 150 respectively. All of theabove phase configurations were obtained for the same set of initial conditions. ε -8-7-6-5-4-3-2 l og ( K ) R (a) ε -8-7-6-5-4-3-2 l og ( K ) R (b) FIG. 7: (color online) The order parameters (a) R corresponding to group 1, and (b) R for group 2 are calculated and plotted between the parameter region0 . < ε < .
0, 10 − < K < − for Ω = .
27. At each of the parameter values we use a random initial condition and iterate the system for 4 × timesteps after which we calculate the order parameters ( R , R , R ) and average them over 10 time steps. The chimera states of case 1 and 2 are found in the regionwhere the subgroup order parameters take values R ≈ , R ≈
0. We have labelled the phase diagram by the final state seen in each region as defined in the text. R , R ε R R (a) R , R log (K)R R (b) FIG. 8: (color online) The variation of the group wise order parameters is plotted for (a) the parameter values K = − , Ω = . , N =
150 as ε variesbetween 0 .
65 and one. (b)The order parameters R and R are plotted as K varies between 10 − and 10 − for the parameter ε = .
93 with Ω = . containing the chimera state STI structures (case 1 and 2). Wenote that the chimera states shown in both Figs. 5 and 7 ex-hibit phase synchronisation in group one and STI structuresin group two. Its clear that due to symmetry in the evolutionEq. 2 discussed earlier in section II, there exists a mirror ver-sion of this chimera where the nature of the dynamics of thegroups are interchanged which can be accessed via a differ-ent set of random initial conditions (see figure 9). The frac-tion of initial conditions which yields each of these symmet-ric configurations are also similar implying equal basin vol-umes (see Fig. 10). In addition to this, Fig. 10 plotted for K = − , Ω = . , ε = . , N =
150 indicates that due tothe multistability of this system (Eq. 2) a fraction of the ini-tial conditions evolve to the other states (case 3 - 5) , withbasins of stability with volume proportional to the fraction inthe histogram.It is useful to examine the basin stability of all the statesobserved in the K − ε space. We examine a 30 ×
30 grid forthe range − < log K < − . < ε <
1, with Ω fixed at0 .
27. At each grid point we choose 400 sets of initial condi-tions with θ values chosen randomly between zero and one.The system in Eq. 2 is then evolved from each of these ini-tial conditions for 4 × time steps to the final state. Thenature of the final state is then identified using the complexorder parameters, R , R , R , which take the specific valuesfor different final states as described in the previous section.Figs 11 and 12 show that all attractors listed in cases 1 - 5have a finite non-zero basin stability in the region bounded by − < log K < − . < ε < . . − . < log K < − . < ε <
1. This fraction is less than 0 . ε < . K < − . K > −
2. InFig.11.b we find that the globally phase desynchronised statehas low basin stability in a significant portion of the parame-ter region of interest and the fraction of initial conditions thatevolve to it less than 0.2 when log K > −
6. However it isnear one when log K < − ε . The twophase clustered state is a favoured state when K > − (seeFig. 12a). The probability for completely random initial con- dition to evolve to a fully synchronised state is however lowin the entire parameter space examined, as can be seen in Fig.12.b. Figs. 13 shows the basin stability of the two mirrorchimera states. It is clear that the two states appear with ap-proximately equal probability in the region of interest in theregion − . < log K < − . < ε <
1. We note thatthe basin volume of the two phase cluster states is of simi-lar magnitude in this region. These figures appear to indicatethe existence of riddling in the basins of attraction of the finalstates. In future we hope to explore this in detail.
V. CHIMERA STATES WITH STI LIKE STRUCTURES INTHE DESYNCHRONISED GROUP
The region in the phase diagram where chimera states withspatiotemporally intermittent behaviour are seen in the K , ε parameter space for Ω = .
27 is clearly identified in the phasediagram of Fig. 7. As mentioned earlier, this is the regionwhere R takes value 1, and R is zero. It can be seen from thespace time plots and the temporal variation of the order pa-rameter R (see Fig. 2) for this chimera state that the maps inthe synchronised group are spatially phase synchronised butthe phase at which they synchronise is not a temporal fixedpoint as shown by the variation of Ψ (Figs. 2.c and f). Thevariation of R , Ψ in Figs. 2.c and 2.(f) maps in the desyn-chronised group can be seen to be spatially and temporallydesynchronised. Here we carry out the linear stability analy-sis of this chimera states for the parameters that where suchsolutions are seen and calculate the Lyapunov exponents. A. Linear stability analysis and Lyapunov spectrum
We find the stability of the chimera states with spatiotem-poral intermittent regions, by calculating the eigenvalues ofthe one step Jacobian matrix. J = (cid:20) A BC D (cid:21) (6)Here A , B , C , D are N × N block matrices which have the form, A = ( + ε N ) f (cid:48) ( θ n ( )) ε N f (cid:48) ( θ n ( )) · · · ε N f (cid:48) ( θ n ( N )) ε N f (cid:48) ( θ n ( )) ( + ε N ) f (cid:48) ( θ n ( )) · · · ε N f (cid:48) ( θ n ( N )) · · · · · · · · · · · · ε N f (cid:48) ( θ N ( )) ε N f (cid:48) ( θ N ( )) · · · ( + ε N ) f (cid:48) ( θ N ( N )) (7) D = ( + ε N ) f (cid:48) ( θ n ( )) ε N f (cid:48) ( θ n ( )) · · · ε N f (cid:48) ( θ n ( N )) ε N f (cid:48) ( θ n ( )) ( + ε N ) f (cid:48) ( θ n ( )) · · · ε N f (cid:48) ( θ n ( N )) · · · · · · · · · · · · ε N f (cid:48) ( θ N ( )) ε N f (cid:48) ( θ N ( )) · · · ( + ε N ) f (cid:48) ( θ N ( N )) (8) B = ε N f (cid:48) ( θ n ( )) ε N f (cid:48) ( θ n ( )) · · · ε N f (cid:48) ( θ n ( N )) ε N f (cid:48) ( θ n ( )) ε N f (cid:48) ( θ n ( )) · · · ε N f (cid:48) ( θ n ( N )) · · · · · · · · · · · · ε N f (cid:48) ( θ n ( )) ε N f (cid:48) ( θ n ( )) · · · ε N f (cid:48) ( θ n ( N )) (9) pha s e map index (a) pha s e map index (b) FIG. 9:
The snapshots of (a) the chimera state where group one are phase synchronised while group two is phase desynchronised and (b) chimera statewhere group one is phase desynchronised and maps in group two is phase synchronised. At parameters Ω = . , K = − , ε = . , N =
150 two differentinitial conditions were used to obtain the above snapshots. B a s i n s t ab ili t y ( i n % ) FIG. 10:
Histogram of the fraction of the initial conditions that evolve to chimera states, two phase clustered state, fully phase desynchronised state andphase synchronised states. We use 400 initial conditions at parameter values K = − , Ω = . , ε = . , N = .
5% of the total phase space while the same measure for the mirrored version is 19%. The basin volume of the fully phasedesynchronised state is 15 .
5% while for the two phase clustered state and fully phase synchronised state they are 46% and 1% respectively of the total phasespace. C = ε N f (cid:48) ( θ n ( )) ε N f (cid:48) ( θ n ( )) · · · ε N f (cid:48) ( θ n ( N )) ε N f (cid:48) ( θ n ( )) ε N f (cid:48) ( θ n ( )) · · · ε N f (cid:48) ( θ n ( N )) · · · · · · · · · · · · ε N f (cid:48) ( θ n ( )) ε N f (cid:48) ( θ n ( )) · · · ε N f (cid:48) ( θ n ( N )) (10)where f (cid:48) ( θ σ n ( i )) = − K cos ( πθ σ n ( i )) and σ = , K takes low values ( < − ) (see the phase diagram in Fig. 7). Using this, and thefact that | cos πθ σ n ( i ) | ≤
1, the quantity f (cid:48) ( θ σ n ( i )) = − K cos 2 πθ σ n ( i ) ≈ ( − α ) where the upper bound on the valueof α is K in the chimera region. Using this approximation, theone step Jacobian matrix takes the form, J c = (cid:20) A BB A (cid:21) (11)where, A = ( − α ) ( + ε N ) ε N · · · ε N ε N ( + ε N ) · · · ε N · · · · · · · · · · · · ε N ε N · · · ( + ε N ) (12) B = ( − α ) ε N ε N · · · ε N ε N ε N · · · ε N · · · · · · · · · · · · ε N ε N · · · ε N (13) Here J c (Eq. A7) is a 2 N × N block circulant matrix whichcan be block diagonalised using a matrix P , P = F ⊗ I N (14)where I N is a N × N identity matrix and F is a 2 × of the form, F = √ (cid:20) ω (cid:21) (15)with ω = exp ( π i ) = cos π + i sin π = −
1. So we have P − J c P = J ∗ c = (cid:20) A + B A − B (cid:21) (16)where A + B = ( − α ) ( + ε + ε N ) ε + ε N ··· ε + ε N ε + ε N ( + ε + ε N ) ··· ε + ε N ··· ··· ··· ··· ε + ε N ε + ε N ··· ( + ε + ε N ) (17) ε -6-5-4-3-2 l og ( K ) ba s i n s t ab ili t y (a) ε -6-5-4-3-2 l og ( K ) ba s i n s t ab ili t y (b) FIG. 11:
The basin stability of (a) chimera phase state (b) fully phase desynchronised state in K − ε space. The parameters Ω = . , N =
150 are keptfixed. ε -6-5-4-3-2 l og ( K ) ba s i n s t ab ili t y (a) ε -6-5-4-3-2 l og ( K ) ba s i n s t ab ili t y (b) FIG. 12:
The basin stability of (a) two phase clustered state (b) fully phase synchronised state in the K − ε space. The parameters Ω = . , N =
150 arekept fixed. A − B = ( − α ) ( + ε − ε N ) ε − ε N ··· ε − ε N ε − ε N ( + ε − ε N ) ··· ε − ε N ··· ··· ··· ··· ε − ε N ε − ε N ··· ( + ε − ε N ) (18) A + B and A − B are block circulant matrices . The j th eigen-value λ j of the matrix A + B and A − B is given by, E A ± Bj = ( − α ) (cid:32) + ε ± ε N + ω j ( ε ± ε ) N + ω j ( ε ± ε ) N + · · · + ω ( N − ) j ( ε ± ε ) N (cid:33) where ω is the N t h root of unity i.e. ω = exp ( π iN ) . Setting j = A + B , A − B . So, E A + B = (cid:18) + ε + ε N × N (cid:19) ( − α )= ( + ε + ε )( − α )= ( − α ) (19) E A − B = (cid:18) + ε − ε N × N (cid:19) ( − α )= ( + ε − ε )( − α )= ε ( − α ) (20)For any j > E A ± Bj = ( − α ) (cid:18) + ε ± ε N ( + ω j + ω j + · · · + ω ( N − ) j ) (cid:19) = − α (21)where we use ε + ε = + ω j + ω j + · · · + ω ( N − ) j =
0. So the eigenvalues of the matrix J ∗ c for K →
0, are 2 ( − α ) , 2 ε ( − α ) and 2 N − − α .1 ε -6-5-4-3-2 l og ( K ) ba s i n s t ab ili t y (a) ε -6-5-4-3-2 l og ( K ) ba s i n s t ab ili t y (b) FIG. 13:
The basin stability of (a) chimera state with phase synchronisation in group one and phase desynchronisation in group two and (b) chimera statewhere the distribution of phases are interchanged between the groups in the K − ε space. The parameters are Ω = . , N = Therefore the Lyapunov exponents are, λ = τ lim τ → ∞ τ ∑ ln 2 ( − α ) ≈ . + ln ( − α ) λ = τ lim τ → ∞ τ ∑ ln ( ε ( − α )) = ln ( ε ) + ln ( − α ) λ j = τ lim τ → ∞ τ ∑ ln ( − α ) = ln ( − α ) for all j = , · · · , N (22)Fig. 14a plots the variation of Lyapunov exponents with ε for K = − . It is clear that the Lyapunov exponents (Eq. 22)match the numerical values obtained from the numerical evo-lution in the chimera regime, i.e. ( − . (cid:46) log K (cid:46) − ) . It isclear from Fig. 14b that λ , λ calculated in Eq. 22 start to de-viate from numerically calculated values when log K > − J (cid:54)≈ J c . We note that two of the Lyapunov expo-nents obtained in range studied, viz λ and λ are positive.Hence the maps in the chimera regime show hyperchaotic be-haviour. We note that chimera state with hyperchaotic tempo-ral dynamics has been observed earlier in coupled oscillatorsystems , where again a hyperchaotic STI chimera has beenseen. We note that this is one of the few instances where thetemporal dynamics of the chimera state with spatiotemporallyintermittent structure is found to be hyperchaotic in nature.The stability analysis shown here applies to all the final statesthat appear in the region of the phase diagram when K is negli-gibly small. The linear stability analysis of globally synchro-nised state (Case 4) and two clustered state (Case 5) is carriedout for arbitrary value of K in appendix A.We examine the temporal dynamics of the chimera statesvia the site return maps by randomly choosing a typical sitefrom each of the groups (see figure 15). We observe that thereis a distinct difference between the return map of a site fromgroup one and group two. The return maps for groups one andtwo show non-banded and banded structures respectively. Thenoninvertible nature of the return map of a site belonging tothe synchronised group can be clearly seen in Fig. 15a. The space time behaviour of the phases of the circle maps in thedesynchronised group suggest the existence of synchronisedislands with identical phases inside clusters of spatiotempo-rally phase desynchronised sites. We analyse this spatiotem-poral intermittent structure of the incoherent group in the nextsection. B. Identifying the laminar and burst sites
We begin the analysis of the spatiotemporally intermittentstructure in the incoherent group of the chimera state by iden-tifying the intermittent synchronised islands (laminar) withinthe desynchronised phases (burst) in the incoherent group.The existence of global coupling between the maps imply thatthe neighbourhood of each map is essentially the entire sys-tem. The coupling terms in the evolution Eq. 2 also show thatphase of any map at a time step depends on phases of all mapsin the system at previous time step. Therefore in order to lo-cate the intermittent synchronised sites we must consider theall the maps at a given time step as well as the previous timestep.We consider any two sites ( i , j ) as laminar sites when thephases of the circle maps at these sites are such that the quan-tity ∆ σ , σ (cid:48) i j = (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) exp ( π i θ σ ( i )) + exp ( π i θ σ (cid:48) ( j )) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) is lessthan an assigned cutoff value set by the parameter δ . Thequantity ∆ σ , σ (cid:48) i j , which can also be considered as a two site or-der parameter (compare with the definition of R , R , R of thegroup-wise and global order parameter given in Eq. 5), is usedinstead of directly computing the phase difference because ∆ σ , σ (cid:48) i j takes into account the fact that equation 2 has a mod-ulo one operation. It is necessary to take account of the globalcoupling topology to identify the laminar and burst sites. Bytaking into account of the global coupling topology, we iden-tify the laminar and burst sites in the spatiotemporal variationof the phases of the CML in two steps which we describe here: 1. We consider the phases of the CML at two consecutive2 L y apuno v e x ponen t s ε λ α ) λ ln(2 ε ) + ln(1 - α ) λ j , j > 2 (a) L y apuno v e x ponen t s log (K) λ α ) λ ln(2 ε ) + ln(1 - α ) λ j , j > 2 (b) FIG. 14:
The variation of the largest Lyapunov exponent λ , the second largest Lyapunov exponent λ and the rest of the LE values are plotted between (a)0 . < ε < K = − and (b) − < log ( K ) < − ε = .
93. In both the case the largest possible value of α = K is used. It can be clearly seen thatthe analytic value ln2 ε + ln ( − α ) matches with the numerically calculated λ . All LE values are calculated via the Gram-Schmidt orthogonalisation methodusing the Jacobian for 10 time steps after a transient of 3 × steps. Other parameters are fixed at Ω = . , N = θ n + θ n (a) θ n + θ n (b) FIG. 15: (color online) The return map of site 10 from (a) group one and (b) group two when the CML is already evolved in to the chimera state atparameters K = − , Ω = . , ε = . , N = time steps, n and n −
1. The phase of the map at site i in group σ at time step n , is denoted as θ σ n ( i ) . Wechoose two sites each from time steps n − n thatcan belong to any of the groups and they are denoted by θ σ n − ( j ) and θ σ (cid:48) n ( i ) . We now check if ∆ σ , σ (cid:48) i j < δ for all i = , , · · · , N for both σ (cid:48) = , θ σ (cid:48) n ( i ) sat-isfies the condition, ∆ σ , σ (cid:48) i j < δ . We also label the latticesite at θ σ n − ( j ) as laminar if at least one such i is foundfor which ∆ σ , σ (cid:48) i j < δ (see Fig. 16.(a) for reference). Werepeat this method for j = , , · · · , N for σ = ,
2. Wethus check if there is any temporal infection betweenthe sites at time step n − n . Once thelaminar sites at time step n are identified by this methodwe check if there is any spatial infection between sites.We describe this in next step.2. Now, we calculate ∆ σ , σ (cid:48) i j = (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) exp ( i πθ σ (cid:48) n ( j )) + exp ( i πθ σ n ( i )) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) for all j = , , · · · , N when σ (cid:48) (cid:54) = σ and for j = , , · · · , N except j (cid:54) = i when σ (cid:48) = σ and we check the condition ∆ σ , σ (cid:48) i j < δ . A simple schematic is shown in Fig. 16.(b)for clarification. We label θ σ n ( i ) as a laminar site attime step n if the condition is satisfied at least once.After checking the phases of the maps at all sites at timestep n for temporal and spatial infections for laminarity in asimilar fashion, we move on to the phases of the maps in thenext time step. The intermittent synchronised and burst sitesin a given spatiotemporal variation of the maps can all be iden-tified in this way. In the next section we find the distributionof laminar and burst segments in the incoherent group of thechimera states. C. Distribution of laminar and burst lengths
We note that the co-evolving maps are placed at consecu-tively numbered sites on a one dimensional lattice, with mapssituated at sites 1 to N being identified as belonging to one3 time = n group one group two time = n ✓ n ( i ) ✓ n ( j ) (a) ✓ n ( i ) time = n group one group two (b) FIG. 16: (color online) A schematic of the method for identifying the laminar and burst sites. Each of these arrows in the above diagrams indicate the pair ofsites between which the condition ∆ σ , σ (cid:48) ij is checked group and maps from N + N being identified as belong-ing to the other subgroup. The maps are coupled globally,with the intra group and intergroup couplings taking distinctvalues. Thus, maps at consecutive sites, are influenced by thebehaviour of all the other maps, with the crucial element beingthe number of maps in each subgroup whose phase angles arein the laminar or burst phase as defined by the pairwise orderparameter ∆ σ , σ (cid:48) i j . It is interesting to see the distribution of lam-inar and burst lengths, viz. the distribution of the lengths ofcoherent and incoherent segments under these circumstances.Here, we find the number of consecutive sites which are lam-inar or burst sites at a given time in the phase desynchro-nised subgroup region during the evolution of chimera states.Thus the length of a laminar ( l L ) /burst ( I B ) segment can varyfrom 0 to N . The probability P ( l L ) for a laminar segmentof length l L to exist during a given time interval is the ra-tio of the total number of laminar segments of all lengths inthis time period. The resulting distribution P ( l L ) of the lami-nar segments as well as the burst segments P ( l B ) is plotted inFigs. 17 and 18, and is seen to follow exponential behaviour α exp ( − β l ) irrespective of system size. Thus, long laminarand burst segments are not very probable. This is unlike thecase of spatiotemporal intermittency in systems with diffusivecoupling where power law distributions are seen for laminarlengths . Table I lists the values of α L , β L for laminar seg-ments and α B , β B for the burst segments for different systemsizes. VI. SIGNATURES OF THE TRANSITION FROM THECHIMERA STATE AND REPRODUCTION OF THE PHASEDIAGRAM
We have noted earlier that the crucial element which gov-erns whether the map at a given site remains in the laminaror burst state, after time evolution from step n to step n + n , as defined bythe pairwise order parameter. This also turns out to be thekey element in the existence of the spatiotemporally intermit-tent chimera. The phase diagram of Fig. 7 which focusses on the chimera region and its boundaries, is constructed us-ing the global order parameter ( R n , and group-wise order pa-rameters R n , R n ) which differentiate between fully phase syn-chronised configurations, partially phase synchronised con-figurations (e.g. chimera states) and fully phase desynchro-nised configurations. This phase diagram, as well as the crosssection taken at log K = − . ε ≈ . m σ which isdefined to be m σ = n (cid:48) n + n (cid:48) ∑ n x σ ( n ) N where x σ ( n ) is the number oflaminar sites at any time step n . If this quantity is plotted asa function of time as in Fig. 19 it can be seen that after an ini-tial transient, m σ settles to the fixed values shown in table II.These are the m σ values in the chimera state and fully phasedesynchronised state.Figs. 20a and 20b show the variation of m σ with ε and K respectively. The variation of m , m clearly indicates the tran-sition from the fully phase desynchronised state to the chimeraphase state in the CML (see Fig. 20.(a)) with increasing val-ues of ε . Here, m , m ≈ .
55 (phase desynchronized val-ues) when ε < .
81 and m ≈ , m ≈ ε > .
81. In fact between the parameter values ε = . .
828 we observe that m = m ≈ ε > .
828 we find that m (cid:46) ε increases to one for this fixed value of K . Comparing Figs. 8.(a) and 20(a) we can see that the quan-tity m σ can differentiate correctly between the chimera with apurely synchronised subgroup and the chimera state with de-fects in the synchronised subgroup. The variation of the orderparameters in Fig. 8.b shows another cross section taken at an-other parameter in the phase diagram, viz. ε = .
93, wheresimilar behaviour is found in the variation of m and m (seeFig. 20.(b)). To compare this through the quantity m σ , wefind that when − . < log K < − . m (cid:46) , m ≈ P(l L ) l L α L exp(- β L l L ) (a) P(l B ) l B α B exp(- β B l B ) (b) FIG. 17:
The probability distribution of (a) laminar ( I L ) and (b) burst ( I B ) segments in the spatiotemporal variation of the phases of incoherent maps in grouptwo in the chimera state. The parameters are K = − , ε = . , Ω = .
27 and N = P ( I L ) ≈ α L exp ( − β L I L ) with α L = . ± . , β L = . ± . P ( I B ) ≈ α B exp ( − β B I B ) where α B = . ± . , β B = . ± . P(l L ) l L α L exp(- β L l) (a) P(l B ) l B α B exp(- β B l) (b) FIG. 18:
The probability distribution of (a) laminar ( I L ) and (b) burst ( I B ) segments in the spatiotemporal variation of the phases of incoherent maps in grouptwo in the chimera state. The parameters are K = − , ε = . Ω = .
27 and N = P ( I L ) ≈ α L exp ( − β L I L ) with α L = . ± . β L = . ± . P ( I B ) ≈ α B exp ( − β B I B ) where α B = . ± . β B = . ± . nised subgroup appear when − . < log K < − .
4, since inthis range of K , m = , m ≈ − < log K < − . < ε < Ω = .
27, using the quantities m and m , i.e. the average frac-tion of laminar sites in groups one and two. It can be seen fromFigures 21.(a) and (b) that the average fraction of laminar sitescorrectly replicates the chimera configuration in the region ap-proximately given by 0 . < ε < − . < log K < − ε values, such that ε > .
8, whereas fully desynchro-nised configurations are seen for values of ε < .
8. Twoclustered states are found between − < log K < − . < ε < ε andfor log K > −
3, both m and m are one, indicating the ex-istence of two clustered states. Within the same range of ε if we decrease K we see that defects start to appear in group one, as m decreases from one. As log K approaches − ε . Forlog K values close to − m to be be comparable to m implying that thechimera configuration is lost. Similarly, the fraction m (cid:46) ε increases from 0 . K is between therange − . − . < m , m < . − . < log K < − ε < . α L , β L , α B and β B for the distribution of laminar and burst segments in the incoherent group of thechimera states for different system sizes. System size ( N ) α L β L α B β B
200 4 . ± . . ± . . ± . . ± . . ± . . ± .
001 0 . ± . . ± . . ± . . ± .
006 1 . ± . . ± . . ± . . ± . . ± . . ± . m , m time(in 50000 steps) m m (a) m , m time(in 50000 steps) m m (b) m , m time(in 50000 steps) (c) FIG. 19: The variation of m and m for during the evolution of (a) case 1 : chimera states with purely synchronised subgroup ( ε = . ) , (b) case 2 : chimera states with defects in the synchronised subgroup ( ε = . ) and (c) case 3 : fully phasesynchronised state ( ε = . ) . Other parameters are kept fixed at K = − , Ω = . N = n (cid:48) = K = − , Ω = . , N =
150 for all three cases below.
Attractor m (numerical) m (numerical)Case 1 a b c a ε = .
82, chimera states with a purely synchronised subgroup b ε = .
93, chimera states with defects in the synchronised subgroup c ε = .
75, fully phase desynchronised state
VII. CONCLUSION
To summarise, we have analysed a system which showsnovel chimera behaviour, viz. a mixed state with a synchro-nised part and a spatiotemporally intermittent part. This be-haviour is seen in a coupled map lattice consisting of twogroups of globally coupled sine circle maps with differentvalues of intergroup coupling and intra-group coupling. Thesystem, when evolved with random initial conditions, showsa variety of solutions in different regions of the parameterspace. A phase diagram is obtained using the complex or-der parameter, and the basin stability of each type of solutionin the context of multiattractor behaviour is discussed. Wenote that the basin stability of the chimera states, is large inthe chimera region, with the chimera and its mirror versionbeing equally probable at all parameter values. We note thatthe STI chimeras are seen at very small values of the nonlin- earity parameter K ( − . (cid:46) K (cid:46) − ) , where the map be-haviour is very close to the behaviour of coupled shift maps.Analytic techniques can be effectively applied in this regime,and confirm the results obtained numerically. The Lyapunovexponent spectrum in this regime is calculated by both meth-ods, and turns out to have two positive exponents, confirmingthat the chimera seen here is a hyperchaotic chimera. We notethat very few examples of hyperchaotic chimeras have beenseen earlier . The parameter values in this regime is simi-lar to the regime where splay chimera states have been seenearlier, with splay initial conditions . However, none of thesplay states observed show hyperchaotic behaviour. One ap-plication context where such low values of K can be realisedis that of coupled Josephson junction arrays with high valuesof capacitance .The spatiotemporally intermittent chimera seen here showsco-existing laminar and burst states, which are identified viaa pairwise order parameter. The distribution of laminar andturbulent lengths drops off exponentially, due to global cou-pling, unlike the power law behaviour seen at some parametervalues for locally diffusive coupling . The global natureof the coupling used here, and the distinct values of inter-group and intragroup coupling, implies that the observed be-haviour is dependent on the number of laminar and turbulentsites in each subgroup. The average fraction of laminar andburst sites saturates to steady state values after an initial tran-sient. This average fraction can be used to construct the phasediagram in the vicinity of the chimera region. This phase di-agram matches exactly the phase diagram constructed via the6 m , m ε m m (a) m , m log K (b) FIG. 20: (color online) (a) The average fractions m , m are calculated at parameters N = , K = − , Ω = .
27. The signature of the transition from thefully phase de-synchronisation to chimera phase state can be seen at ε = .
8. Here the value of m becomes one while m remains less than one as expectedfor the chimera phase state in the CML. (b) The variation of the average fraction m and m are plotted as log K varies between − − ε fixedat 0 .
93 with Ω = .
27. Chimera states with defects in the synchronised group appear when log K < − .
27 while chimera states with purely phasesynchronised group appear between − . < log K < − .
4. When log K is between − .
36 and − . K . The system settles to the two clustered state when log K > − . ε -8-6-4-2 l og ( K ) m (a) ε -8-6-4-2 l og ( K ) m (b) FIG. 21: (color online) The average fraction of laminar sites (a) m and (b) m are calculated over 10 time after discarding initial 3 × time steps at eachparameter between − < log K < −
2, 0 . < ε < Ω = .
27 and N = order parameter, confirming that the average number of lami-nar and turbulent sites is the crucial factor in the spatiotempo-ral dynamics of the chimera. A cellular automaton with globalcoupling which incorporates these features can be easily con-structed. We hope to explore this approach in future work,and examine its consequences for the analysis of the chimerastate. We also hope to explore the consequences of the hyper-chaotic behaviour seen in the chimera state seen here, and itsimplications for experimental systems such as coupled lasermodels and Josephson junction arrays where such chimerascan be realised. Appendix A: Linear stability analysis of the globallysynchronised state and the two phase clustered state1. The globally synchronised state
The analysis of the globally synchronised state has beencarried out in Ref. . We summarise this over here. In order tocarry out the linear stability analysis for the globally synchro-nised state, θ σ n ( i ) = θ sync , ∀ σ = , i = , , , · · · N attime step n , the one step Jacobian matrix (Eq. 6) is convertedto a block circulant form using a similarity transformation viaa matrix given by a direct product of 2 × and an N × N identity matrix. The transformed Jacobian isgiven as,7 C + D = ( − K cos2 πθ sync ) ( + ε + ε N ) ε + ε N ··· ε + ε N ε + ε N ( + ε + ε N ) ··· ε + ε N ··· ··· ··· ··· ε + ε N ε + ε N ··· ( + ε + ε N ) (A1) C − D = ( − K cos2 πθ sync ) ( + ε − ε N ) ε − ε N ··· ε − ε N ε − ε N ( + ε − ε N ) ··· ε − ε N ··· ··· ··· ··· ε − ε N ε − ε N ··· ( + ε − ε N ) (A2) The eigenvalues λ j of the matrix C + D and C − D is given by, E C ± Dj = ( − K cos 2 πθ sync ) (cid:32) + ε ± ε N + ω j ( ε ± ε ) N + ω j ( ε ± ε ) N + · · · + ω ( N − ) j ( ε ± ε ) N (cid:33) (A3)where ω is the N t h root of unity i.e. ω = exp ( π iN ) . Setting j = C + D , C − D . So, E C + D = ( − K cos 2 πθ sync ) (A4) E C − D = ε ( − K cos 2 πθ sync ) (A5)For any j > E C ± Dj = ( − K cos 2 πθ sync ) (A6)where we use ε + ε = + ω j + ω j + · · · + ω ( N − ) j =
0. So the eigenvalues of the matrix J ∗ c for K →
0, are2 ( − K cos 2 πθ sync ) , 2 ε ( − K cos 2 πθ sync ) and 2 N − − K cos 2 πθ sync . The eigenvalues ofthe Jacobian matrix for the shift map case K = ε and 2 N −
2. Two clustered state
Using the fact that the phases in group one take the values θ n ( i ) = θ and those in group two take the values θ n ( i ) = θ for all i = , , , · · · N −
1, we find the eigenvalues of the Ja-cobian matrix in Eq. 6. We verify the eigenvalue spectrumby calculating the upper bound on the largest eigenvalue us-ing the Gershgorin theorem analytically and checking if theentire eigenvalue spectra is less than the upper bound as dis-cussed in Ref. . The Jacobian matrix in this case is givenby, J c = (cid:20) E FG H (cid:21) (A7) Here E , F , G , H are N × N block matrices which have the form, E = ( + ε N ) f (cid:48) ( θ ) ε N f (cid:48) ( θ ) · · · ε N f (cid:48) ( θ ) ε N f (cid:48) ( θ ) ( + ε N ) f (cid:48) ( θ ) · · · ε N f (cid:48) ( θ ) · · · · · · · · · · · · ε N f (cid:48) ( θ ) ε N f (cid:48) ( θ ) · · · ( + ε N ) f (cid:48) ( θ ) (A8) H = ( + ε N ) f (cid:48) ( θ ) ε N f (cid:48) ( θ ) · · · ε N f (cid:48) ( θ ) ε N f (cid:48) ( θ ) ( + ε N ) f (cid:48) ( θ ) · · · ε N f (cid:48) ( θ ) · · · · · · · · · · · · ε N f (cid:48) ( θ ) ε N f (cid:48) ( θ ) · · · ( + ε N ) f (cid:48) ( θ ) (A9) F = ε N f (cid:48) ( θ ) ε N f (cid:48) ( θ ) · · · ε N f (cid:48) ( θ ) ε N f (cid:48) ( θ ) ε N f (cid:48) ( θ ) · · · ε N f (cid:48) ( θ ) · · · · · · · · · · · · ε N f (cid:48) ( θ ) ε N f (cid:48) ( θ ) · · · ε N f (cid:48) ( θ ) (A10) G = ε N f (cid:48) ( θ ) ε N f (cid:48) ( θ ) · · · ε N f (cid:48) ( θ ) ε N f (cid:48) ( θ ) ε N f (cid:48) ( θ ) · · · ε N f (cid:48) ( θ ) · · · · · · · · · · · · ε N f (cid:48) ( θ ) ε N f (cid:48) ( θ ) · · · ε N f (cid:48) ( θ ) (A11)where f (cid:48) ( θ σ ) = − K cos 2 πθ σ and σ = , J c are real and nonnegative which implies that the Gershgorinrow region and the column region will consist of disks whosecenters lie on the real axis. For the two phase clustered statethe centre, ( c j ) of the j th Gershgorin disk is given by, c σ j = (cid:0) + ε N (cid:1) ( − K cos 2 πθ σ ) , σ = , for j = { , , · · · , N − } and σ = j = { N , N + , · · · , N } . The radius of the j th8 -1-0.5 0 0.5 1 0 0.5 1 1.5 2 I m g ( λ ) Re( λ ) Gershgorin diskseigenvaluesupper bound
FIG. 22:
Eigenvalues and the Gershgorin disks of the Jacobian matrix ofthe system calculated for the two clustered state. The parameters are ε = . , Ω = . , K = − , N = disc in the Gershgorin row region r j is, r j = ε ( N − ) N ( − K cos θ ) + ε ( − K cos θ ) for j = , , , · · · , N − r j = ε ( N − ) N ( − K cos θ ) + ε ( − K cos θ ) for j = N , N + , · · · , N (A12)The radius of the i th disc in the column region is s i = (cid:18) ε ( N − ) N + ε (cid:19) ( − K cos θ ) for j = { , , , · · · , N − } s i = (cid:18) ε ( N − ) N + ε (cid:19) ( − K cos θ ) for j = { N , N + , · · · , N } (A13)Since the centres of every disc in the Gershgorin row and col-umn region lie on the real axis, the two bounds set by the Ger-shgorin row and column regions are given by the two largestnumbers at which the discs from each of these sets intersectthe real axis i.e. max ( c σ j + r σ j ) and max ( c σ i + s σ i ) for σ = , i , j = , , · · · , N −
1. The required bound on the eigen-values is the minimum of these two values. So the upperbound on the eigenvalues of Jacobian for the two clusteredstate is, min (cid:0) max (cid:0) c σ j + r σ j (cid:1) , max ( c σ i + s σ i ) (cid:1) for σ = , i , j = , , , · · · , N −
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