Clustered chimera states in delay coupled oscillator systems
aa r X i v : . [ n li n . PS ] M a r Clustered chimera states in delay coupled oscillator systems
Gautam C. Sethia ∗ and Abhijit Sen Institute for Plasma Research, Bhat, Gandhinagar 382 428, India
Fatihcan M. Atay
Max Planck Institute for Mathematics in the Sciences, Leipzig 04103, Germany
We investigate chimera states in a ring of identical phase oscillators coupled in a time-delayedand spatially non-local fashion. We find novel clustered chimera states that have spatially dis-tributed phase coherence separated by incoherence with adjacent coherent regions in anti-phase.The existence of such time-delay induced phase clustering is further supported through solutionsof a generalized functional self-consistency equation of the mean field. Our results highlight anadditional mechanism for cluster formation that may find wider practical applications.
PACS numbers: 05.45.Xt, 89.75.Kd
The study of time delay induced modifications in thecollective behaviour of systems of coupled nonlinearoscillators is a topic of much current interest both for itsfundamental significance from a dynamical systems pointof view and for its practical relevance to modeling ofvarious physical, biological and chemical systems. In reallife situations time delay is usually associated with finitepropagation velocities of information signals, latencytimes of neuronal excitations, finite reaction times ofchemicals etc. and in collective oscillator studies it is usu-ally modeled through a time delayed coupling function.Many such past model studies on globally and locallycoupled oscillator systems have uncovered interestingand sometimes novel time delay induced modificationsof the equilibrium, stability and bifurcation propertiesof their collective states[1] . While global and local(nearest neighbour) coupling models have traditionallyreceived much attention there is now a growing interestin the collective dynamics of models with non-localcouplings [2, 3, 4, 5, 6, 7]. Non-local coupling canbe relevant to a variety of applications such as in themodeling of Josephson junction arrays [8], chemicaloscillators [5, 6, 9], neural networks for producing snailshell patterns and ocular dominance stripes [10, 11, 12]etc. One of the striking features of non-locally coupledoscillator systems is that they can support an unusualcollective state in which the oscillators separate into twogroups - one that is synchronized and phase locked andthe other desynchronized and incoherent [5]. Such astate of co-existence of coherence and incoherence doesnot occur in either globally or locally coupled systemsand has been named as a chimera state by Abrams &Strogatz [13]. The nature and properties of this exoticcollective state as well as its potential applications arestill not fully explored or understood and thereforecontinue to offer exciting future possibilities. It is notknown for example whether such chimera states canexist in the presence of time delay in the system andif so then what their characteristics are. This is theprincipal question we examine in this work through numerical simulations and mathematical analysis ofa model system consisting of a ring of densely anduniformly distributed identical phase oscillators thatare coupled in a time-delayed and spatially non-localfashion. We find that chimera states do indeed existbut acquire an additional spatial modulation such thatthe single spatially connected phase coherent region ofthe usual chimera state is now replaced by a numberof spatially disconnected regions of coherence withintervening regions of incoherence. Furthermore theadjacent coherent regions of this clustered chimera stateare found to be in anti-phase relation with respect toeach other. To understand the origin and the nature ofthis pattern we have extended the mean field approachused by Kuramoto [5] and applied it to our systemwhich has a distance-dependent time delay factor in thecoupling and have derived a functional self-consistencyequation. A numerical solution of this self-consistencyequation yields a space-dependent order parameter anda space-dependent mean phase function that confirm theexistence and explain the nature of the spatial patternof the oscillator phases.We consider the following model equation representingthe continuum limit of a chain of identical phase oscilla-tors arranged on a circular ring C , ∂∂t φ ( x, t ) = ω − Z L − L G ( x − x ′ ) × sin[ φ ( x, t ) − φ ( x ′ , t − τ x,x ′ ) + α ] dx ′ (1)where 2 L is the system length, ω is the natural frequencyof the oscillator and a closed chain configuration is en-sured by imposing periodic boundary conditions. Thekernel G ( x − x ′ ), appropriately normalized to unity overthe system length, is taken as, G ( x − x ′ ) = k − e − kL ) e − kd x,x ′ (2)which provides a non-local coupling among the oscilla-tors over a finite spatial range of the order of k − whichis taken to be less than the system size. The coupling istime delayed through the argument of the sinusoidal in-teraction function, namely, the phase difference betweentwo oscillators located at x and x ′ is calculated by takinginto account the temporal delay for the interaction signalto travel the intervening geodesic (i.e. shortest) distancedetermined as d x,x ′ = min {| x − x ′ | , L − | x − x ′ |} . Thetime delay term is therefore taken to be of the form, τ x,x ′ = d x,x ′ /v where v is the signal propagation speed.In the absence of time delay the above equation reducesto the one investigated in [5, 6]. The constant phase shiftterm α in the undelayed model breaks the odd symme-try of the sinusoidal coupling function and as discussedin [13, 14] it is needed as a tuning parameter for ob-taining chimera solutions in the undelayed case. In thepresence of time delay however we find that α no longerplays such a critical role since the time delay factor alsofulfills a similar function.We now describe direct numerical simulation resultsobtained by solving Eq.(1) using a large number of dis-crete oscillators (typically N=256). The set of systemparameters chosen for the simulations illustrated herewere, 2 L = 1 . α = 0 . k − = 0 . ω = 1 . v = 0 . τ max ) in the system of 5 .
12. As discussed in past stud-ies [5, 6], the choice of appropriate initial conditions isvery important for numerically accessing a chimera state.Kuramoto used a random distribution with a Gaussianenvelope for the initial distribution of the phases to ob-tain a chimera solution. For our time-delayed systemwe find that choosing the initial phases of the oscillatorsfrom a uniform random distribution between 0 and 2 π and then arranging them in a mirror symmetric distri-bution in space provides a rapid access to a clusteredchimera state. Our simulations have been done with theXPPAUT [15] package using a Runge-Kutta solver (witha small integration time step of δt = 0 .
01) till a timestationary solution is obtained and tested for indepen-dence from discreteness effects by repeating the runs for N = 128, 256 and 512. In panels (a) and (b) of Fig.1we show a space time plot of our simulation for the pa-rameters mentioned above in the early stages of evolu-tion (starting from random initial phases) and in the fi-nal stages of the formation of a clustered chimera staterespectively. Panel (c) shows a snapshot of the spatialdistribution of the phases in the final stationary state.We see four coherent regions interspersed by incoherenceand also note that the adjacent coherent regions are inanti-phase. Panel (d) is a blowup of the region between x = − . x = − .
25 giving an enlarged view of anincoherent region and portions of the adjacent coherentregions. These solutions are also found to be quite robustand show no signs of instability over arbitrarily large in-tegration times. We have tested the integrity of the solu-tions for times well over 100 τ max . A detailed parametricstudy of the stability regions is presently under progress T i m e T i m e −0.5 −0.25 0 0.25 0.5230240250−0.5 −0.25 0 0.25 0.505 φ −0.5 −0.45 −0.4 −0.35 −0.3 −0.2505 φ x (a) (b) (c) (d) FIG. 1: (color online) (a) The space-time plot of the oscillatorphases φ for the parameters 2 L = 1 . k = 4 .
0, 1 /v = 10 . ω = 1 . α = 0 . x = − . x = − .
25 giving an enlarged view of an incoherent regionand portions of the adjacent coherent regions. and will be reported elsewhere.To gain a better understanding of the nature of thispattern and of the dynamics of its formation we havecarried out a mathematical analysis based on the gener-alized mean field concept as developed by Kuramoto forthe non-delayed case. For this we first rewrite Eq.(1) interms of a relative phase θ ( x, t ) = φ ( x, t ) − Ω t (where Ωrepresents a rotating frame in which the dynamics simpli-fies as much as possible such that with the phase-lockedportions rotate with this constant drift frequency) as, ∂∂t θ ( x, t ) = ω − Ω − Z L − L G ( x − x ′ ) × sin[ θ ( x, t ) − θ ( x ′ , t − τ x,x ′ ) + α + Ω τ x,x ′ ] dx ′ (3)The key idea behind Kuramoto’s analysis of chimerastates was the introduction of a mean field like quan-tity, namely, a complex order parameter Re i Θ , defined ina manner analogous to what is done for globally coupledsystems. For our case we write, R ( x, t ) e i Θ( x,t ) = Z L − L G ( x − x ′ ) e i [ θ ( x ′ ,t − τ x,x ′ ) − Ω τ x,x ′ ] dx ′ (4)The above order parameter differs from the usual defi-nition for global coupling systems in several ways - thespatial average of e iθ is weighted by the coupling kernel G ( x − x ′ ), the phase θ is evaluated in a time delayedfashion and the factor e − i Ω τ x,x ′ adds a complex phase tothe kernel G ( x − x ′ ). The latter two features provide afurther generalization of Kuramoto’s analysis carried outfor a non-delayed system [5, 6, 13, 14].In terms of R and Θ, Eq.(1) can be rewritten as : ∂∂t θ ( x, t ) = ∆ − R ( x, t ) sin[ θ ( x, t ) − Θ( x, t ) + α ] (5)where ∆ = ω − Ω. Eq.(5) is in the form of a singlephase oscillator equation being driven by a force termwhich in this case is the mean field force. To obtain astationary pattern (in a statistical sense) we require R and Θ to depend only on space and be independent oftime. Under such a circumstance the oscillator popu-lation can be divided into two classes: those which arelocated such that R ( x ) > | ∆ | can approach a fixed pointsolution ( ∂θ ( x, t ) /∂t = 0) and the other oscillators thathave R ( x ) < | ∆ | would not be able to attain such anequilibrium solution. The oscillators approaching a fixedpoint in the rotating frame would have phase coherentoscillations at frequency Ω in the original frame whereasthe other set of oscillators would drift around the phasecircle and form the incoherent part. Following the pre-scription provided by Kuramoto [5, 6] for the undelayedcase, we substitute the solutions of Eq.(5) for the twoclasses of oscillators into the integrand on the right handside (R.H.S.) of Eq.(4) and obtain the following func-tional self-consistency condition, R ( x ) e i Θ( x ) = e iβ Z L − L G ( x − x ′ ) e i [Θ( x ′ ) − Ω τ x,x ′ ] × ∆ − p ∆ − R ( x ′ ) R ( x ′ ) dx ′ (6)where β = π/ − α . We need to solve for three unknowns— the functions R ( x ), Θ( x ) and the quantity ∆. Con-dition (6) provides only two equations when we separateits real and imaginary parts. A third condition can beobtained by exploiting the fact that the equation is in-variant under any rigid rotation Θ( x ) → Θ( x ) + Θ . Wecan therefore specify the value of Θ( x ) at any arbitrarychosen point, e.g. Θ( L ) = 0. We have solved Eq.(6)numerically by following a three step iterative procedureconsisting of the following steps. We choose arbitrary butwell behaved initial guess functions for R ( x ) and Θ( x )and use the condition Θ( L ) = 0 in one of the equationsof (6) to obtain a value for ∆. The initial profiles and the∆ value so obtained are used to evaluate the R.H.S. of(6) to generate new profiles for R and Θ. These are nextused to generate a new value of ∆ and the procedure isrepeated until a convergence in the value of ∆ and thefunctions R and Θ are obtained.A MATHEMATICA program incorporating this algo-rithm was developed and benchmarked against the re-sults for the no-delay case. Fig.2 shows the rapid and ex-cellent convergence in ∆ to a unique value of ∆ = 0 . R ∆ Iteration number
FIG. 2: Variation of ∆ with the iteration number showinga rapid convergence in the numerical solution of the self-consistency Eq.(6). The system parameters are identical tothose used in the direct solution of Eq.(1). and Θ) are shown in Fig.3 and the converged value of∆ is marked in the upper panel by the horizontal line.The amplitude of the order parameter ( R ) shows a peri-odic spatial modulation - peaking at four symmetricallyplaced spatial locations. The corresponding phases ofthe order parameter are seen to be in anti-phase for ad-jacent peaks in R . In between the peaks R is seen todip to very small values at certain locations such that R ( x ) < | ∆ | which should correspond to the incoherentdrifting parts of the chimera. To better appreciate theagreement between the direct solutions of Eq.(1) and themean field solutions of Eq.(6) we have plotted the resultstogether in Fig.4. As is clearly seen the measured orderparameter ( R and Θ) and ∆ from the direct simulationsof Eq.(1) match well with the results of solving Eq.(6).The spatial profile of the phases ( φ ) of the oscillators asobtained from the direct simulation of Eq.(1) is shownin the top panel of Fig.4. We see four coherent regionsinterspersed by incoherence as expected from the resultsof solving Eq.(6).We note from Fig.(3) and Fig.(4) that both R andΘ are mirror symmetric (i.e. R ( x ) = R ( − x ) , Θ( x ) =Θ( − x )), a property that the original phase Eq.(1) alsopossesses. Eq.(1) is also invariant under the transforma-tion ( φ ( x, t ) → − φ ( x, t ) , ω → − ω, α → − α ) and can havesolutions with such a symmetry as well, namely, travel-ing wave solutions given by φ ( x, t ) = Ω t + πqx/L . Inour numerical simulations we find that by changing theinitial conditions, but keeping the same system parame-ters, we can also get traveling wave solutions. There alsoseems to be a clear correspondence between the numberof clusters of the observed chimera state and the wavenumber q of the co-existent traveling wave solution. Forthe 4-cluster chimera of Fig.3 the co-existent travelingwave has q = 2 and similar results have been obtainedfor 6-cluster ( q = 3) and 8-cluster ( q = 4) chimera solu-tions. −0.5 −0.25 0 0.25 0.500.20.4R x−0.5 −0.25 0 0.25 0.50123 Θ x FIG. 3: Spatial profiles of the amplitude R and the phase Θ ofthe order parameter obtained by solving the self-consistencyEq.(6) by an iterative scheme. The horizontal line in theupper panel, marks the converged value ∆ = 0 . −0.5 −0.25 0 0.25 0.50246 φ −0.5 −0.25 0 0.25 0.500.5 R −0.5 −0.25 0 0.25 0.50123 Θ x −0.5 −0.25 0 0.25 0.5−0.200.2 ω − φ x (a) (b) (c) (d) . FIG. 4: (color online) (a) The phase pattern for a clusteredchimera state as obtained by direct simulation of Eq.(1). Themeasured spatial profiles of the order parameter ( R and Θ)from these simulations are shown in panels (b) and (c) asdashed curves and compared with the solutions from the self-consistency Eq.(6) shown as solid curves. (d) ω − ˙ φ for theoscillators from a direct simulation of Eq.(1). The horizontallines in (b) and (d) mark the converged value of ∆ = 0 . To conclude, we have demonstrated for the first timethe existence of chimera type solutions in a time-delayedsystem of non-locally coupled identical phase oscillators.Time delay is found to lead to novel clustered states witha number of spatially disconnected regions of coherencewith intervening regions of incoherence. The adjacent co-herent regions of this clustered chimera state are foundto be in anti-phase relation with respect to each other.Our numerical simulations are further validated and ex-plained through solutions of a generalized functional self-consistency equation of the mean field. The mean fieldparameters (the amplitude and phase of the complex or-der parameter) clearly reflect the modulated nature ofthe effective driving force on each oscillator and lead to the resultant pattern of phase distribution seen in theclustered chimera state. Thus time delay offers an ad-ditional mechanism for cluster formations in dynamicalsystems and model systems incorporating time delay mayprovide a useful paradigm for studying this phenomenon.Our results can be usefully extended to higher dimen-sions e.g. to examine the influence of time delay on spi-ral wave based chimeras in two dimensions [9] and mayalso help provide insights into experimental observationsof clustered states that are generic to many chemical andbiological systems[16].GCS thanks P. Abbott and S. Richardson for help onMathematica. FMA and GCS acknowledge the hospital-ity of MPI-PKS, Dresden, Germany, at DYONET 2006. ∗ Electronic address: [email protected][1] H. G. Schuster and P. Wagner, Prog. Theor. 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