Chimeras with uniformly distributed heterogeneity: two coupled populations

Carlo R. Laing [email protected] School of Natural and Computational Sciences, Massey University, Private Bag 102-904 North Shore Mail Centre, Auckland, New Zealand

Abstract

Chimeras occur in networks of two coupled populations of oscillators when the oscillators in one population synchronise while those in the other are asynchronous. We consider chimeras of this form in networks of planar oscillators for which one parameter associated with the dynamics of an oscillator is randomly chosen from a uniform distribution. A generalisation of the approach in [C.R. Laing, Physical Review E, 100, 042211, 2019], which dealt with identical oscillators, is used to investigate the existence and stability of chimeras for these heterogeneous networks in the limit of an infinite number of oscillators. In all cases, making the oscillators more heterogeneous destroys the stable chimera in a saddle-node bifurcation. The results help us understand the robustness of chimeras in networks of general oscillators to heterogeneity.

Chimera states, Coupled oscillators, Bifurcations, Collective behavior in networks, Synchrony

I Introduction

Chimera states occur in networks of coupled oscillators and are characterised by coexisting groups of synchronised and asynchronous oscillators [panabr15, ome18]. They have been observed in one-dimensional [abrstr06, omezak15] and two-dimensional [lai17, panabr15a] domains with nonlocal coupling, and also networks formed from two populations with strong coupling within a population and weaker coupling between them [abrmir08, pikros08, panabr16]. A variety of oscillator types have been considered, with the most common being a phase oscillator [kurbat02], but others include Stuart-Landau oscillators [lai10], van der Pol oscillators [omezak15, uloome16], oscillators with inertia [boukan14, olm15] and neural models [olmpol11, omeome13, ratpyr17].

Many investigations of chimeras report only the results of numerically solving a finite number of ordinary differential equations (ODEs) which describe the networks’ behaviour. Such simulations are for only a finite time, so the results seen may actually be part of a long transient [Zakkap16]. With a finite network there is the issue of finite size effects, such as positive Lyapunov exponents which tend to zero as the network size is increased [ome18] or chimeras’ finite lifetimes [wolome11]. Perhaps most significantly, such simulations cannot detect unstable states so it is often not clear what happens to a stable chimera as a parameter is varied, other than it no longer existing.

Early results on the existence of chimeras used a self-consistency approach [lai10, kurbat02, shikur04, abrstr04] but this does not provide information on the stability of solutions. A great deal of progress has been made using the Ott/Antonsen ansatz [ottant08, ottant09], since it gives evolution equations for quantities of interest, but its use is restricted to networks of phase oscillators coupled through sinusoidal functions of phase differences [abrmir08, lai09, ome18]. Laing [lai10] used self-consistency to investigate the existence of chimeras in networks of two populations of Stuart-Landau oscillators, each oscillator being described by a complex variable. This was later generalised [lai19] using techniques from [clupol18] to determine the stability of these chimeras, and chimeras in networks of three more types of oscillators (Kuramoto with inertia, FitzHugh-Nagumo oscillators, delayed Stuart-Landau oscillators) were studied.

The approach in [lai19] was to recognise that the incoherent oscillators in one population lie on a curve $\mathcal{C}$ in the phase plane while those in the synchronous population can be described by a pair of real variables, since all of these oscillators are identical and undergo the same dynamics. In the limit of an infinite number of oscillators in each population the curve $\mathcal{C}$ is described by its shape (distance from the origin in polar coordinates) and the density of oscillators on it, and partial differential equations (PDEs) governing the evolution of these functions can be derived [clupol18]. The full network is then described by a pair of PDEs and a pair of ODEs, coupled by an integral.

The analysis in [lai19] assumed identical oscillators, but we do not expect this to be the case in any experimental situation [tinnko12, marthu13, totrod18] and it is known that networks of identical oscillators may have qualitatively different dynamics from those of nonidentical oscillators [watstr94]. In this paper we extend the results in [lai19] to the case of nonidentical oscillators. Specifically, we assume that for each oscillator, one parameter associated with its dynamics is randomly chosen from a uniform distribution. A uniform distribution is zero outside some range and this means that for a narrow distribution, the types of chimeras observed in [lai19] persist and can be described by a generalisation of the techniques developed in that paper.

Various distributions of intrinsic frequencies in networks of all-to-all coupled oscillators have been considered, e.g. Lorentzian [ottant08], bimodal [marbar08], Gaussian [strmir91, hanfor18], beta [daeds20] and uniform [piedes18, paz05, bairos10, ottstr16, eydwol17]. There are significant differences in the transition to synchrony as coupling strength is increased between distributions with compact support and those whose support is unbounded. In the former case one normally observes a first-order transition, whereas in the latter it is second-order. Also, for an infinite network full synchrony — in which all oscillators are phase locked — can only occur when the frequency distribution has compact support [erm85]. We observe and exploit this phenomenon to analyse the networks studied in this paper.

Previous relevant work includes [rybvad19], which considers a network formed from coupled ring subnetworks of logistic maps in which a number of parameters are randomly chosen from uniform distributions. The authors investigate the effects of varying the widths of these distributions on the number of subnetworks which fully synchronise.

We consider networks formed from two populations of oscillators. In Sec. II we consider Kuramoto-type phase oscillators and in Sec. LABEL:sec:SL we revist the Stuart-Landau oscillators studied in [lai10]. Sec LABEL:sec:pend considers Kuramoto oscillators with inertia, also studied in [lai10]. We study van der Pol oscillators in Sec. LABEL:sec:vdp and conclude in Sec. LABEL:sec:disc.

II Kuramoto phase oscillators

We first consider two populations of phase oscillators coupled through a sinusoidal function of phase differences. Networks of this form have been studied previously [abrmir08, lai09a, panabr16, pikros08, monkur04]. We first consider heterogeneity in intrinsic frequencies, then in the strength of coupling between populations.

II.1 Distributed frequencies

Consider two populations of $N$ phase oscillators each governed by

\frac{d\theta_{j}}{dt}=\omega_{j}+\frac{\mu}{N}\sum_{k=1}^{N}\sin{(\theta_{k}-\theta_{j}-\alpha)}+\frac{\nu}{N}\sum_{k=1}^{N}\sin{(\theta_{N+k}-\theta_{j}-\alpha)}

(1)

for $j=1,2\dots N$ and

\frac{d\theta_{j}}{dt}=\omega_{j}+\frac{\mu}{N}\sum_{k=1}^{N}\sin{(\theta_{N+k}-\theta_{j}-\alpha)}+\frac{\nu}{N}\sum_{k=1}^{N}\sin{(\theta_{k}-\theta_{j}-\alpha)}

(2)

for $j=N+1,N+2,\dots 2N$ . $\mu$ is the strength of coupling within a population and $\nu$ is the strength between populations. For identical $\omega_{j}$ this system reduces to the system studied in [abrmir08, panabr16, pikros08] while if they are chosen from a Lorentzian distribution it is the same as in [lai09a]. Instead, here for each population the $\omega_{j}$ are randomly chosen from the uniform distribution $p(\omega)$ which is non-zero only on the interval $B$ .

An example of a chimera state for (1)-(2) is shown in Fig. LABEL:fig:snapkur where $p(\omega)$ is uniform on $[-\Delta\omega,\Delta\omega]$ . We see that population 1 is incoherent, with no apparent dependence of $\theta_{j}$ on $\omega_{j}$ , whereas population 2 is synchronised (although not phase synchronised) with a clear dependence of $θ Missing Operator$