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Multiscale dynamics in communities of phase oscillators

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10.1063/1.3672513

### Abstract

We investigate the dynamics of systems of many coupled phase oscillators with heterogeneous frequencies. We suppose that the oscillators occur in *M* groups. Each oscillator is connected to other oscillators in its group with “attractive” coupling, such that the coupling promotes synchronization within the group. The coupling between oscillators in different groups is “repulsive,” i.e., their oscillation phases repel. To address this problem, we reduce the governing equations to a lower-dimensional form via the ansatz of Ott and Antonsen, Chaos **18**, 037113 (2008). We first consider the symmetric case where all group parameters are the same, and the attractive and repulsive coupling are also the same for each of the *M* groups. We find a manifold of neutrally stable equilibria, and we show that all other equilibria are unstable. For *M* ≥ 3, has dimension *M* − 2, and for *M* = 2, it has dimension 1. To address the general asymmetric case, we then introduce small deviations from symmetry in the group and coupling parameters. Doing a slow/fast timescale analysis, we obtain slow time evolution equations for the motion of the *M* groups on the manifold . We use these equations to study the dynamics of the groups and compare the results with numerical simulations.

© 2012 American Institute of Physics

Received 12 August 2011
Accepted 06 December 2011
Published online 03 January 2012

Lead Paragraph:

The dynamics of hierarchically structured networks is particularly relevant to systems with very large numbers of dynamical units. Examples can be found in neuroscience and in the study of social dynamics in large populations. At the coarsest level, one can often think of such systems as being based on interactions between distinct groups or communities of dynamical network nodes. Thus, it is of great interest to investigate the kinds of dynamical behaviors that can result when network systems have interacting communities and to develop techniques that may be useful for studying such systems. In this paper, we address these issues for the illustrative, paradigmatic case in which the nodal dynamics is describable within the phase oscillatormodel.

Acknowledgments: This work was supported by an NSF REU grant to the University of Maryland and by ONR Grant No. N00014-07-1-0734. Also thanks to user ghazwozza of the social media website reddit.com for contributing to the geometric “tip-to-tail” formulation of the *S* = 0 equilibria in Sec. III B.

Article outline:

I. INTRODUCTION

II. LOW DIMENSIONAL FORMULATION

III. IDENTICAL GROUPS

A. Equilibria

B. Equilibria with *S* = 0

C. Equilibria with *S* ≠ 0

D. Numerical simulations

E. Stability of equilibria

1. Equilibria with S = 0 and r_{σ} ≠ 0 for all σ

2. Equilibria with S = 0 and one or more incoherent groups

3. Equilibria with S ≠ 0

IV. NONIDENTICAL GROUPS

A. Formulation

B. The examples of M = 3 and M = 4

C. Numerical results

V. CONCLUSION

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/content/aip/journal/chaos/22/1/10.1063/1.3672513

2012-01-03

2014-04-24

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