Coupled oscillator systems:Chimera states

Bhargava Ravoori

May 14, 2008

Abstract

Arrays of coupled identical oscillators can display interesting spatio-temporal patterns in which domains of coherent, phase locked oscillatorsexist alongside incoherent and desynchronized oscillators. This behaviorhas been observed in a variety of systems drawn from various disciplinesof science. Recently, studies have been done to incorporate the effects ofdelay into the model equations as well. Here I review some of the recentliterature devoted to the examination of such states while attempting toreproduce a part of the results.

1 Introduction

The dynamics of arrays of coupled limit-cycle oscillators is relevant to severalbranches of science including physics, chemistry, neural sciences and biology. Insuch systems, the possibility of various dynamical states depends sensitively onthe nature of coupling mechanisms. More particularly, the range of couplingamong the group of oscillators is seen to have telling effects on the systemdynamics. The coupling range may be broadly classified to be either local, globalor intermediate. Local coupling describes the scenario where each oscillator iscoupled with ones in its close neighborhood while global coupling refers to acase in which each oscillator is coupled with every other oscillator in the system.Intermediate coupling deals with scenarios where the coupling strength dropsoff as the distance between oscillators increases. Such a coupling scheme iscommonly refered to as nonlocal coupling in literature.

In the case of nonlocal coupling, systems of oscillators have been observedto exist in exotic spatio-temporal states in which domains of coherent, phaselocked oscillators co-exist with inchoherent and drifting oscillators. Abramsand Strogatz christened such a state, chimera, after the Greek mythologicalcreature with the head of a lion, the body of a goat and a serpents tail. Theexistence of chimera states was first observed in a ring of phase oscillators [1]and was later analytically studied further [2].In this paper, I will discuss aboutthe various systems in which chimera states have been observed. In section 2,I will present examples of chimera states drawn from various fields of sciencealong with some of the results that I reproduced. In section 3, I will discussthe incorporation of delay in the non-locally coupled phase oscillator model.

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Finally, I will summarize the current state of research on chimera systems andcomment on possible future work.

2 Chimera states: Examples

2.1 Reaction-diffusion systems

The chimera state has first been observed in chemical reaction-diffusion systemsby Battagtokh and Kuramoto [1]. The dynamics of a reaction-diffusion systemare given by the equation

∂x∂t

= F(x, µ) + D.∇2x (1)

Here, the first term represents the reaction dynamics while the second termrepresents diffusive coupling. mu is a parameter of the system.

Certain chemical systems show interesting oscillatory behavior in terms ofthe concentrations of the chemical species involved. We consider such a systemand assume that it breaks into an oscillatory state at µ = 0 . That is to say,the system undergoes a Hopf bifurcation as the parameter µ is varied abovezero. If the frequency of this temporal frequency is denoted ω,then any physicalquantity in the vicinity of this transition can be described as

C(x, t) = Co + A(x, t)eiωot + A∗(x, t)e−iωot + (higher order terms) (2)

where ωo is the temporal frequency at threshold and A describes the complexamplitude of the physical quantity. It is a common assumption that the am-plitude varies very slowly close to the bifurcation point and thus the higherorder terms may be neglected. It is derived in [3, 4] that the behavior of theamplitude, A, is governed by the complex Ginzburg Landau equation (CGLE).

∂A∂t

= (1 + ico)A− (1 + ic2)|A2|A + α∇2A (3)

where α is genrally complex. For simplicity, we choose to treat the problem inone spatial dimension. The 1-D CGLE in the presence of non-local coupling isderived in [5] and is written as

∂A

∂t= (1 + ico)A− (1 + ic2)|A2|A+K(1 + ic1)(B(x, t)−A(x, t)) (4)

whereB(x, t) =

∫dx′G(x− x′)A(x′, t) (5)

In [1], Battagtokh and Kuramoto consider the kernel in the integrand to be

G(x) =12e−κ|x| (6)

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The authors solve equation (4) numerically on a finite interval [0, 1] usingperiodic boundary conditions. The initial condition is assumed such that theamplitude of oscillations, |A| = 1 everywhere. They find that for suitable valuesof parameters in the weak coupling regime(K << 1) , the system devolops achimera state. The resultant phase pattern is shown in figure 1(a). Clearly,the pattern has two distinct spatial domains. Close to the boundaries is aregion with coherent phase oscillation while in the central region, the spatialcontinuity in phase is completely lost. Further, since the coupling strength,K, is assumed to be small, all the spatially distributed oscillators attain themaximum amplitude (i.e. they behave nearly like uncoupled oscillators) andhence the amplitude pattern is not of interest. It is also seen that all theoscillators exhibit an approximate collective oscillation with a constant driftfrequency denoted Ω. They then proceed to rewrite equation (4) as a phaseequation which is much easier to analyze. This equation is obtained using thephase reduction method described in [4].

∂φ(x, t)∂t

= ω −∫dx′G(x− x′) sin(φ(x, t)− φ(x′, t) + α) (7)

where the constants are related by

tan(α) =c2 − c11 + c1c2

(8)

Simulating this equation with the same parameter values produces a spatio-temporal state similar to the results from equation (4) (Figure 1(b)). Thusthey show that rewriting equation (4) as a phase equation gives an equivalentalternative way of looking at the dynamics of the system. Now, defining arelative phase

ψ(x, t) = φ(x, t)− Ωt (9)

Equation 7 may be rewritten as

∂ψ(x, t)∂t

= ω − Ω−∫dx′G(x− x′) sin(ψ(x, t)− ψ(x′, t) + α) (10)

Introducing a complex order parameter with a modulus R and phase Θ,∫dx′G(x− x′)eiψ(x′,t) = R(x, t)eiΘ(x,t) (11)

we will have

∂ψ(x, t)∂t

= ω − Ω−R(x, t)sin(ψ(x, t) + α+ Θ(x, t) ) (12)

In this equation, R may be interpreted as the strength of a forcing field drivingthe phase. The time averaged spatial profiles of R(x, t) and Θ(x, t) are shown inFigures 1(c),1(d). The forcing mean field amplitude is seen to be stronger nearthe boundaries explaining why the oscillators near the boundary are entrained

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to be in step with each other. The field is weak near the center causing theoscillators to move out of coherence. Further, it is seen that the frequencyof the oscillators near the boundaries equals the drift frequency Ω, which isto be expected as an effect of the strong forcing mean field amplitude, whilethe smooth nature of R(x) near the center implies a smooth distribution offrequencies near the center.The authors then treat the problem analyticallyand establish a theoretical footing for the observations [1]. In [2], Abrams andStrogatz treat the same phase equation problem with a different kernel

G(x) =1

2π(1 +Acos(x)) (13)

0 ≤ A ≤ 1

which they show numerically, gives qualitatively similar solutions. They thentreat the problem analytically and derive closed form solutions for R and Θ [2].

(a) (b)

(c) (d)

Figure 1: (a) Spatial phase profile of the CGLE solution, (b) Spatial phaseprofile of the reduced phase oscillator equation solution, (c) The order parametermodulus R, (d) The order parameter phase, Θ

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2.2 Reaction-diffusion systems: A 2-D example

In [6], Shima and Kuramoto extend the above analysis to a 2D system of coupledFitzHugh-Nagumo oscillators whose equations may, again, be reduced to anequivalent phase equation using the phase reduction method. They use thekernel

G(r) =1

2πr2o

Ko(r

ro), r = |r| (14)

where Ko is the modified Bessel function of the second kind. They find thatthe system exhibits spiral waves with a phase randomized core in the case ofweak coupling while exhibiting a spatially coherent spiral wave in the stronglimit case as shown in Figure 2.

Figure 2: Spiral patterns exhibited by nonlocally coupled FitzHugh-Nagumooscillators for two representative cases of strong coupling ( (a),(b), and(c) )and weak coupling ( (d), (e), and (f) ). Overall patterns are displayed in grayscale. (b) and (e) show their structure near the core. (c) and (f) show thecorresponding phase portraits.

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2.3 Reaction-diffusion systems : CGLE with forcing

The case of a CGLE with non-local coupling and parametric external forcinghas been considered by Kawamura in [8]. The equation describing such a systemis given as

∂A

∂t= (1 + ico)A− (1 + ic2)|A2|A+K(1 + ic1)(B(x, t)−A(x, t)) + γA∗ (15)

Here γ denotes the strength of external forcing. Note that in the limit as γtends to zero, the equation reduces to a non-locally coupled CGLE consideredbefore. Here, the author uses the kernel

G(x) =12e−|x| (16)

which is the same as equation (6) with κ = 1.To reproduce the results presented in [8], I solve equation 15 numerically for

the real and imaginary parts of A. In solving the equation, I impose Neumannboundary conditions while considering a spatial grid of 2048 points spaced ∆x =.005 apart. The origin is assumed to be at the center of the grid. The resultsfor the case of strong coupling are presented in Figure 3. Figure 3(a) showsthe phase portrait while Figure 3(b) and Figure 3(c) show the spatial phaseand amplitude profiles respectively. Figure 4 shows the results for the case ofweak coupling. It is clearly evident that the chimera state tends to form in thecase of weak coupling only. Further, it is to be noted that in both the strongand the weak coupling case K

γ is held constant at 1.5. If we disregard B inequation (15) treating it as an external forcing, the linearized equation for thecomplex amplitude at the center,x = 0, is given by

∂A

∂t= (1 + ico)A−K(1 + ic1)A+ γA∗ (17)

A linear stability analysis of the stationary solution A gives the eigen values

λ± = 1−K ±√γ2 − (co −Kc1)2 (18)

Therefore the necessary condition for a chimera state to appear, i.e. the condi-tion for the oscillator at the center to drift is expressed as λ± > 0. Clearly thecondition is satisfied if K < 1 which is the case of weak coupling.

2.4 Nonlocally coupled neural oscillators

Synchronization of neural oscillators plays a crucial role in several functionsof the brain. Even so, synchronization among neurons is not always desirableas it is the cause of several neurological disorders such as Parkinson’s diseaseand epilepsy. The stability of the synchronized solutions depends on the detailsof the model equations of the neural oscillators. The response time or thedelay parameter is an important parameter in determining the stability of the

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(a) (b) (c)

Figure 3: Solution of CGLE with forcing. ( Strong coupling, K = 1.5 ). (a)Phase portrait, (b) Spatial phase profile (c) Spatial modulus profile

(a) (b) (c)

Figure 4: Solution of CGLE with forcing. ( Weak coupling, K = 0.06 ). (a)Phase portrait, (b) Spatial phase profile (c) Spatial modulus profile

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solution. In [7], the author studies the instability of the synchronized solutionin a system of nonlocally coupled Hodgkin-Huxley equations for the neurons.An exponentially decaying kernel of the form in equation (6) is used. He findsnumerically that for a certain set of parameters, the synchronous state becomesunstable and the sytem settles in a chimera state. Again, he finds that all theoscillators close to the boundary oscillate coherently while the ones in the centerare randomized in phase.

3 Effect of delay in phase oscillator systems

Delay induced effects in various complex systems is a topic of recent intensestudy. In real life situations, delays are caused by various factors like finitepropagation velocities of signals, latency of neuron excitations, finite chemicalreaction times etc...and are unavoidable. It is thus important to include theeffects of delay in any reasonably practical model. Thus far we have consideredseveral cases of nonlocally coupled oscillator systems but none of the modelsincorporate the effects caused due to time delays. The effect of delay in a systemof nonlocally coupled phase oscillator systems has been studied in [9]. In thepresence of time delay, the equation for a chain of phase oscillators becomes(comapre with equation (7))

∂φ(x, t)∂t

= ω −∫ L

−Ldx′G(x− x′) sin(φ(x, t)− φ(x′, t− τx,x′) + α) (19)

with the kernel modified to being

G(x− x′) =k

2(1− e−kL)e−kdx,x′ (20)

The oscillators are considered to be arranged on a circular ring (ensuredby the imposition of periodic boundary conditions) of length 2L. We note that,here, the kernel is normalized to unity over the system length while in the earliercases it was normalized over (−∞,∞). Here, k−1 determines the spatial rangeover which the oscillators are coupled and is taken to be less than the length ofthe system. The coupling is time delayed through the argument of the sinusoidalinteraction function, i.e. the phase difference between the oscillators at x andx′ . The delay is calculated by taking into account the propagation time of theinteraction signal along the shortest distance between x and x′. Thus τx,x’ iscalculated as

τx,x′ =min|x− x′|, 2L− |x− x′|

v(21)

where v is the signal propagation speed.The authors use a spatial grid with 256 points with the length of the system

being 2L = 1. Using an Runge-Kutta solver, they solve for the phase of theoscillators until they settle to a roughly constant spatial profile. The results they

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obtain are shown in Figure 3. It is interesting to see that with the introductionof delay multiple phase randomized domains form. I believe the number ofphase randomized domains seen in a unit interval is closely related to the rangeof interaction ( k−1 ). There are four interaction lengths in the length of thesystem as k = 4. Also, while in the earlier models (equation 10) the parameterα served to break the odd symmetry of the sinusoidal coupling function, herethe delay already fulfills this function. A similar theoretical analysis to thatdone in [1, 2] is done with the incorporation of delay in the definition of theorder parameter. It is found that the numerical results match the analyticalpredictions well.

Figure 5: (a) The space time plot of the oscillator phases φ for the parameters2L = 1.0, k = 4.0, 1/v = 10.24, ω = 1.1, α = 0.9 in the early stages of theevolution. (b) A later time evolution. (c) A snapshot of the final stationarystate. (d) A blowup of the region between x = −0.5 to x = −0.25, giving anenlarged view of an incoherent region and portions of the adjacent coherentregions.

4 Conclusions

Nonlocal coupling in a continuous spatial field of identical oscillators leads tothe formation of interesting spatio-temporal states that are not seen in thecase of local or global coupling schemes. Work on analyzing such states hasbeen relatively recent. Nonlocal coupling is the natural form of coupling mostcommonly seen in practice and a better understanding of this form of couplingmight give us insights into pattern formation in general.It has also been seenthat incorporating time delays in the coupling mechanism would lead to havingmultiple domains of incoherent oscillators. It would interesting to see whateffect time delays might have in other systems like the cases of the 2D reactiondiffusion system and the neural oscillator system.

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References

[1] Y. Kuramoto, D. Battagtokh, Nonlinear Phenom. Complex Syst. 5,380(2002).

[2] L. D. M. Abrams, S. H. Strogatz, Phys. Rev. Lett. 93, 174102(2004).

[3] M. Ipsen, F. Hynne, P. G. Sorensen,Physica D 136, 66(2000).

[4] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Dover, NewYork, 2003).

[5] D. Tanaka, Y. Kuramoto, Phys. Rev. E 68, 026219 (2003).

[6] S. I. Shima, Y. Kuramoto, Phys. Rev. E 69, 036213 (2004).

[7] H. Sakaguchi, Phys. Rev. E 73, 031907 (2006).

[8] Y. Kawamura, Phys. Rev. E 75, 056204 (2007).

[9] G. C. Sethia, A. Sen, Phys. Rev. Lett 100, 144102 (2008).

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