Research paper
Chimeras in multivariable coupled Rössler oscillators

https://doi.org/10.1016/j.cnsns.2020.105661Get rights and content

Highlights

  • Chimera states can be created in an ensemble of multivariable coupled chaotic oscillators.

  • By tuning the coupling parameter in one variable, the region of multistability can be shifted. This provides an additional means to create chimera states.

  • The coexistence of stable synchronised dynamics with desynchronised motion is indicated by the Master Stability Function.

Abstract

We study the coexistence of synchronous as well as asynchronous dynamical behaviours namely chimera states in an ensemble of nonlinear oscillators coupled through different variables. In this system, such states are a result of multistability induced by the coupling in one variable. By tuning the coupling parameter in a different variable, the region of multistability can be shifted. This provides an additional means to create chimera states. We employ this technique in an ensemble of coupled Rössler systems where we observe that there are multiple attractors and the associated basins are intertwined. For such systems, the strength of incoherence (SI) is a useful order parameter through which chimera states can be effectively characterized. The coexistence of stable synchronized dynamics with desynchronized motion is indicated by the master stability function, which we compute for different attractors.

Introduction

The phenomenon of synchronization has been of great interest in the past several decades [1], [2], with numerous studies being devoted to the transition from desynchronized to synchronized dynamics in various branches of the physical, biological, and social sciences [1]. There has been a surge of interest in chimera states [3], [4], [5], [6], [7], [8], [9], [10], namely coexisting groups of synchronized, coherent dynamics, along with incoherent, asynchronous dynamics in ensembles of identically coupled identical oscillators. Such symmetry-breaking was first reported in an ensemble of coupled phase oscillators with nonlocal couplings [3], [4] but subsequently, dynamical chimeras have also been seen in a large variety of other instances. Recent work [11] has discussed brain chimeras whereas the idea of pacemaker oscillator was used to control these states [12].

A large number of theoretical studies have shown that chimera states can occur in different scenarios such as systems with time delay coupling [13], [14], [15], in networks with nonlocal coupling of variable range [16], [17], [18], and in systems with induced multistability, namely the creation of multiple coexisting attractors as a consequence of interactions. In such a case, a chimera state can emerge as a result of the multistability due to the coupling that effectively drives the system parameters to a regime of multistable dynamics [9]. In an ensemble of identically coupled identical chaotic oscillators, one can find complex chimera states, but these will disappear when the coupling is removed. Chimera states have been shown to occur in networks of Stuart-Landau oscillators [19], while chimera-like states have been reported in a network of Rössler oscillators under the influence of attractive and repulsive global coupling [20]. The effect of the intensity parameter in the emergence of chimera states for globally coupled Rössler oscillators was studied [8] and characterized with the help of quantitative measures [21].

Experimental studies have also confirmed the stability of such states in systems ranging from ensembles of metronomes [22], [23] to chemical oscillators [7], and globally coupled networks of semiconductor lasers with delayed optical feedback [24].

In Ref. [17], authors have studied the transition from coherence to incoherence in logistic map and Rössler system. In case to Rössler system, they have considered oscillators that are coupled uniformly in all the three variables x,y and z. Depending upon the coupling strength and coupling radius, the authors have observed spatial patterns and have studied the transition from coherence to incoherence leading to spatial chaos. In the present work, our interest is in the emergence of chimeric dynamics in networks of oscillators that are coupled through more than one variable, and in particular, through pairs of couplings that have opposing effects in the context of synchronization. The motivation for such a study arises from the fact that in practical applications, the nodes of a given network may engage in more than one type of interaction, and one means of exploring this is to study the dynamics of networks where nodes interact with each other through more than one connection [25], [26], [27], [28], [29]. Moreover, in contrast to Ref. [17], we do not consider uniform coupling strength namely, coupling in both the variables may be different. Another important aspect is the sign of coupling constants. While most studies have focussed on the positive values of coupling strengths, a natural generalization would be to allow the coupling strength to have both values. Based on the models of spin glasses [30], several authors have unraveled the effect of random negative and positive coupling. For positive values of coupling strengths, oscillators tend to fall in line with collective rhythm whereas for the negative value of coupling strengths, the oscillators are repelled by the prevailing rhythms [31], [32].

We consider the specific model of the Rössler oscillator in three dimensions with evolution equationsx˙=yz,y˙=x+ay,z˙=b+z(xc),that are linear except for a single nonlinear term in the velocity term in the z direction. In the present work we consider ensembles of coupled Rössler flows. In Section 2 below we first explore the dynamics of two Rössler oscillators coupled in two variables with different coupling strengths. Each of the couplings individually induces multistability in the system, and thus varying the relative strengths of the couplings offers an additional means to create chimera states. We then extend this, in Sections 3 and 4, to the case of an ensemble of interacting chaotic oscillators having nonlocal and global interactions respectively. In such a situation one can exploit the occurrence of multistability to create chimera states in the ensemble of coupled oscillators. The extent of the chimera region can be tuned by varying the coupling in second variable. This technique is fairly general in the sense that emergence of a robust chimera state is not a consequence of time-delay [13] or inherent multistability in the system [9], but results from induced multistability [33], [34]. In Section 5, we study the stability of the synchronized state through the master stability function. This is followed in Section 6 by a discussion and summary.

Section snippets

Coupled Rössler oscillators

Consider a pair of identical Rössler oscillators coupled as followsx˙1=y1z1,y˙1=x1+ay1+ε1(y2y1),z˙1=b+z1(x1c)+ε2(z2z1),x˙2=y2z2,y˙2=x2+ay2+ε1(y1y2),z˙2=b+z2(x2c)+ε2(z1z2).The systems are diffusively coupled via both the y1,2 and the z1,2 variables, and by adjusting the strengths ε1 or ε2 individually or together, the two systems can be made to synchronize.

The Rössler system has rich dynamics, and for the calculations reported here we choose the parameter values a=b=0.1,c=9.0 when

Ensemble of non-locally coupled Rössler oscillators

Consider a ring of N non-locally coupled Rössler oscillators with linear diffusive coupling. The dynamical equations are given byx˙i=yizi,y˙i=xi+ayi+ε12pj=ipj=i+p(yjyi),z˙i=b+zi(xic)+ε22pj=ipj=i+p(zjzi),where, i=1,2,,N label the oscillators and we chose the system parameters to be a=0.1, b=0.1 and c=9. The coupling strength in variables yi and zi is given by ε1 and ε2 respectively. If there are N oscillators in the ensemble, then p is the number of nearest neighbours on each side of

Effect of global coupling

As a special case of Eq. (8), we consider globally coupled oscillators whose equations of motion can be written asx˙i=yizi,y˙i=xi+ayi+ε1(1N1jiyjyi),z˙i=b+zi(xic)+ε2(1N1jizjzi),where, a=0.1, b=0.1 and c=9 and ε1 and ε2 are the coupling parameters. Note that these equation can be obtained by considering r=0.5 in Eq. (8). This again reduces to the form of effectively driven system given by Eq. (9) where f¯yi(t)=1N1jiyj and f¯zi(t)=1N1jizj.

The general features enunciated here are

Local stability of the synchronous states

To determine the stability of the synchronous states, we apply the formalism of the Master Stability Function (MSF) that can be calculated based only on the knowledge about the dynamics of individual oscillators and the coupling function [42], [55]. A typical network of N coupled oscillators can be written as dxidt=F(xi)εj=1NGijH(xj), where F represents the dynamics of uncoupled system, H(x) is a coupling function, ε is a global coupling parameter, and G is a coupling matrix determined by the

Summary

In this work, we consider Rössler oscillators on a ring coupled non-locally in variables yi and zi. We observe that such a population of coupled oscillators leads to the emergence of dynamical chimeras. This is possible in the absence of explicit nonuniformity in the coupling, without time-delay and without multistability in the uncoupled systems. In an ensemble of Rössler oscillators where coupling is either global or nonlocal, by tuning the coupling constants one can create chimeras with

CRediT authorship contribution statement

Anjuman Ara Khatun: Software, Data curation, Visualization, Investigation, Writing - review & editing. Haider Hasan Jafri: Conceptualization, Methodology, Writing - original draft, Supervision, Validation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

AAK acknowledges UGC, India for the financial support. HHJ would like to thank the UGC, India for the award of grant no. F:30-90/2015 (BSR). We also thank Ram Ramaswamy and Awadhesh Prasad for useful discussions.

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