^{1,a)}

### Abstract

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.

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 two

^{1,11}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 populations

^{13}as we vary the nature of the nonlocal coupling among the populations.

This research was supported in part by NSF (Grant Nos. DMS-0412757 and CCF 0835706). I would like to thank Steve Strogatz for helpful discussions and advice throughout the scope of this project, and F. Schittler-Neves, G. Bordyugov, and A. Pikovsky for valuable discussions.

I. INTRODUCTION

II. GOVERNING EQUATIONS

A. Reduced equations and symmetry manifolds

III. BIFURCATION BEHAVIOR NEAR THE TRIANGULAR STRUCTURE

IV. LIMITS OF EXISTENCE FOR CHIMERAS

V. DISCUSSION

### Key Topics

- Oscillators
- 35.0
- Bifurcations
- 20.0
- Networks
- 12.0
- Manifolds
- 7.0
- Coupled oscillators
- 6.0

## Figures

Network structures resulting from varying parameter . The gray disks symbolize populations, inhabited by individual oscillators symbolized by black dots. Their bidirectional coupling is represented by black lines. Each population has a self-coupling of unit strength 1. The population in what becomes the center for is coupled to the neighboring populations with strength ; the populations to the left and right are coupled with strength . The case of a triangular network is obtained for ; the character of the network has chain-like character for .

Network structures resulting from varying parameter . The gray disks symbolize populations, inhabited by individual oscillators symbolized by black dots. Their bidirectional coupling is represented by black lines. Each population has a self-coupling of unit strength 1. The population in what becomes the center for is coupled to the neighboring populations with strength ; the populations to the left and right are coupled with strength . The case of a triangular network is obtained for ; the character of the network has chain-like character for .

Effect of breaking the rotational symmetry on the bifurcation diagram for the SDS and DSD symmetries. The triangular case (see Ref. 13 ) with is shown in the left column in (a) and (c) for comparison with the case of broken symmetry in the right column in (b) and (d). The displayed curves are the saddle-node curve (solid red), the Hopf curve (dashed blue), and the homoclinic curve (dotted black). Dots mark the bifurcation points obtained by inspection of the phase plane. The homoclinic curve is an interpolation based on these points, whereas the solid curves were obtained analytically.

Effect of breaking the rotational symmetry on the bifurcation diagram for the SDS and DSD symmetries. The triangular case (see Ref. 13 ) with is shown in the left column in (a) and (c) for comparison with the case of broken symmetry in the right column in (b) and (d). The displayed curves are the saddle-node curve (solid red), the Hopf curve (dashed blue), and the homoclinic curve (dotted black). Dots mark the bifurcation points obtained by inspection of the phase plane. The homoclinic curve is an interpolation based on these points, whereas the solid curves were obtained analytically.

Bifurcation diagram for the SDS chimera (above) and the two DSD chimeras (below) in the -plane for a range of values. Saddle node (solid red) and Hopf curves (dashed blue) are shown. The curves related to the SDS chimera collapse onto the -axis at ; conversely, the two DSD curves collapse on the axis at and , respectively. It is seen that the Bogdanov–Takens point (black dots) of the upper DSD chimera follows the associated saddle-node curve as we vary .

Bifurcation diagram for the SDS chimera (above) and the two DSD chimeras (below) in the -plane for a range of values. Saddle node (solid red) and Hopf curves (dashed blue) are shown. The curves related to the SDS chimera collapse onto the -axis at ; conversely, the two DSD curves collapse on the axis at and , respectively. It is seen that the Bogdanov–Takens point (black dots) of the upper DSD chimera follows the associated saddle-node curve as we vary .

Boundaries for the occurrence of saddle-node transitions in the -plane (i.e., saddle-node curves at ) shown for the SDS symmetry (a) and the two DSD symmetries in (b) [ is the boundary for the second DSD state seen in the upper part in Fig. 3(b) ]. The regions shaded in gray either have negative coupling or have coupling without chain-like character, as explained in the text. Throughout the regions labeled with No chimeras, we find no chimeras. The black dots indicate points of special interest. (A) is where the saddle-node curves detach from the origin -plane. (B) are the points for which the BT points collide with the -axis leading to the annihilation of the chimera state; , , and . (C) Intersection with boundaries of positive coupling; and . (D) Intersection of the saddle-node boundary with ; . The regions where chimera exists for , which are stable within the symmetry manifolds SDS and DSD, are hatched green.

Boundaries for the occurrence of saddle-node transitions in the -plane (i.e., saddle-node curves at ) shown for the SDS symmetry (a) and the two DSD symmetries in (b) [ is the boundary for the second DSD state seen in the upper part in Fig. 3(b) ]. The regions shaded in gray either have negative coupling or have coupling without chain-like character, as explained in the text. Throughout the regions labeled with No chimeras, we find no chimeras. The black dots indicate points of special interest. (A) is where the saddle-node curves detach from the origin -plane. (B) are the points for which the BT points collide with the -axis leading to the annihilation of the chimera state; , , and . (C) Intersection with boundaries of positive coupling; and . (D) Intersection of the saddle-node boundary with ; . The regions where chimera exists for , which are stable within the symmetry manifolds SDS and DSD, are hatched green.

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