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Chaos / Volume 22 / Issue 1 / REGULAR ARTICLES

Chaos 22, 013102 (2012); http://dx.doi.org/10.1063/1.3672513 (12 pages)

Multiscale dynamics in communities of phase oscillators

Dustin Anderson1, Ari Tenzer2, Gilad Barlev3, Michelle Girvan3, Thomas M. Antonsen3, and Edward Ott3

1Department of Physics and Astronomy, Carleton College, Northfield, Minnesota 55057, USA
2Department of Physics, Washington University in St. Louis, St. Louis, Missouri 63105, USA
3Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA

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(Received 12 August 2011; accepted 6 December 2011; published online 3 January 2012)

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 math of neutrally stable equilibria, and we show that all other equilibria are unstable. For M ≥ 3, math 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 math. We use these equations to study the dynamics of the groups and compare the results with numerical simulations.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. LOW DIMENSIONAL FORMULATION
  3. IDENTICAL GROUPS
    1. Equilibria
    2. Equilibria with S  = 0
    3. Equilibria with S  ≠ 0
    4. Numerical simulations
    5. 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
  4. NONIDENTICAL GROUPS
    1. Formulation
    2. The examples of M = 3 and M = 4
    3. Numerical results
  5. CONCLUSION

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KEYWORDS and PACS

Keywords

chaos, oscillators, synchronisation

PACS

  • 05.45.Xt

    Synchronization; coupled oscillators

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3672513

PUBLICATION DATA

ISSN

1054-1500 (print)  
1089-7682 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    J. A. Acebron, L. L. Bonilla, C. J. P. Cincente, R. Ritort, and R. Spigler, Rev. Mod. Phys. 77, 137 (2005).

    E. Barreto, B. R. Hunt, E. Ott, and P. So, Phys. Rev. E 77, 036107 (2008).


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