Chimera and globally clustered chimera: Impact of time delay

arXiv:1003.4345v1 [nlin.AO] 23 Mar 2010 Chimera and globally clustered chimera: Impact of time delay Jane H. Sheeba,1 V. K. Chandrasekar,1 and M. Lakshmanan1 1Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli - 620 024, Tamilnadu, India Following a short report of our preliminary results [Phys. Rev. E 79, 055203(R) (2009)], we present a more detailed study of the effects of coupling delay in diffusively coupled phase oscillator populations. We find that coupling delay induces chimera and globally clustered chimera (GCC) states in delay coupled populations. We show the existence of multi-clustered states that act as link between the chimera and the GCC states. A stable GCC state goes through a variety of GCC states, namely periodic, aperiodic, long– and short–period breathers and becomes unstable GCC leading to global synchronization in the system, on increasing time delay. We provide numerical evidence and theoretical explanations for the above results and discuss possible applications of the observed phenomena. PACS numbers: 05.45.Xt, 2.30.Ks, 89.75.-k, 87.85.dq I. INTRODUCTION Kuramoto, Battogokh and Shima discovered [1–3] an interesting spatiotemporal pattern which was later named chimera by Abrams and Strogatz [4, 5]. The name chimera, which literally refers to something that is composed of seemingly incompatible or incongruous parts, was coined for this phenomenon because a group of identical oscillators splits into two groups of completely different character. Since its discovery [1, 2, 4], vari- ous theoretical and numerical developments have been reported on the stability of chimera states and their ex- istence in systems with varied structures [4, 6], including time delay [7]. It is clear that the chimera state cannot be attributed to partial synchronization. The occurrence of partial synchronization in a population of non-identical oscillators is not surprising. On the other hand, if an identical group of oscillators splits into synchronized and desynchronized groups, it is called chimera. Therefore, the discovery of chimera came as a surprise in the study of synchronization phenomenon in complex systems. By and large, synchronization in coupled oscillator sys- tems has been analytically and numerically investigated in a rigorous manner over the past years [9, 10]. Possible routes to global synchronization and methods to control synchronization have also been proposed [11, 12]. How- ever, complete understanding of the effects induced by coupling delay in synchronization of coupled oscillator systems is still an open problem. It is well known that time delay occurs in real physical systems. For example, in a network of neuronal populations, there is certainly a significant delay in propagation of signals. In addition there can also be synaptic and dendritic delays. Other examples include finite reaction times of chemicals and fi- nite transfer times associated with the basic mechanisms that regulate gene transcription and mRNA translation. The nature of coupling in coupled oscillator systems has been conventionally considered as instantaneous dur- ing earlier studies. One of the main reasons for this as- sumption is that it substantially simplifies the analysis of 0 50 100 0 3 6 t θi (1,2) Chimera 0 100 200 0 3 6 Globally clustered chimera FIG. 1: (Color online) Occurrence of stable chimera and GCC states in system (1). Black and green (grey) lines represent os- cillators in the first and the second populations, respectively. Here {f, h} = {sin(θ), cos(θ)}, τ1 = nτ2 = nτ with n = 0, A = 0.4, B = 0.6 and τ = 2 for the chimera and τ = 4 for the GCC. the system. In addition, such an approximation is more often physically justified. However, the fact is that the consideration of time delay is vital for modeling real life systems. Furthermore, as we will demonstrate in this pa- per, certain interesting dynamical phenomena in complex systems are characteristic of time delay and they will not occur in systems without time delay. Since the intro- duction of time delay increases the effective dimension of the system, one can expect certain complex phenomena to be explained in a better way in models of real physical systems when delay is included. In this paper, following our Rapid Communication [13], we present a more detailed discussion of the effects of coupling delay in inducing chimera and globally clus- tered chimera (GCC) states in systems of coupled iden- tical oscillator populations. By a GCC state, here we mean a state where the system splits into two different groups, one synchronized and the other desynchronized, each group comprised of oscillators from both the popu- lations. Since a global clustering (mixing) of oscillators from both the populations occur in this case, we call this state a GCC. This is different from the chimera state 2 Population−I Population−II FIG. 2: Schematic representation of system (1) with N = 3 comprised of two populations of all–to–all cou

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