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Chimeras in a network of three oscillator populations with varying network topology

Erik A. Martens
Max Planck Research Group for Biological Physics and Evolutionary Dynamics, Max Planck Institute for Dynamics and Self-Organization, Göttingen 37073, Germany
(Received 24 May 2010; accepted 21 September 2010; published online 24 November 2010; publisher error corrected 2 December 2010)
“We study a network of three populations of coupled phase oscillators with identical frequencies. The populations interact nonlocally, in the sense that all oscillators are coupled to one another, but more weakly to those in neighboring populations than to those in their own population. Using this system as a model system, we discuss for the first time the influence of network topology on the existence of so-called chimera states. In this context, the network with three populations represents an interesting case because the populations may either be connected as a triangle, or as a chain, thereby representing the simplest discrete network of either a ring or a line segment of oscillator populations. We introduce a special parameter that allows us to study the effect of breaking the triangular network structure, and to vary the network symmetry continuously such that it becomes more and more chain-like. By showing that chimera states only exist for a bounded set of parameter values, we demonstrate that their existence depends strongly on the underlying network structures, and conclude that chimeras exist on networks with a chain-like character.
Lead Paragraph
Collective synchronization of coupled oscillators is a problem of fundamental importance and occurs in a wide range of systems including Josephson junction arrays, circadian pacemaker cells in the brain, or the metabolism of yeast cells.22 , 25 , 27 The Kuramoto model has been studied under the influence of global (all-to-all) and local (nearest neighbor) coupling in great detail.4 , 8 The intermediate case of nonlocal coupling, where the coupling strength decays with distance in a network, was first investigated by Kuramoto et al.9 In 2002 they observed a remarkable novel state where a population of identical oscillators splits into two subpopulations, one being synchronized and the other desynchronized, which is called chimera state.10 Since then, several studies have been concerned with its bifurcation behavior and its emergence under the aspects of heterogeneous oscillator frequencies or delayed coupling.1 , 12 , 18 , 23 Chimeras have been observed on a variety of network structures such as rings,2 , 3 networks with two1 , 11 and three oscillator populations,13 and two dimensional (2D) lattices in the shape of spiral waves.15 , 24 A natural question arises: which network topologies allow for the existence of chimeras?17 Here we determine for the first time the limits for the existence of chimeras in a simple network of three oscillator populations13 as we vary the nature of the nonlocal coupling among the populations.