The dynamics of n weakly coupled identical oscillators

P. Ashwin, James W Swift

Research output: Contribution to journalArticle

156 Citations (Scopus)

Abstract

We present a framework for analysing arbitrary networks of identical dissipative oscillators assuming weak coupling. Using the symmetry of the network, we find dynamically invariant regions in the phase space existing purely by virtue of their spatio-temporal symmetry (the temporal symmetry corresponds to phase shifts). We focus on arrays which are symmetric under all permutations of the oscillators (this arises with global coupling) and also on rings of oscillators with both directed and bidirectional coupling. For these examples, we classify all spatio-temporal symmetries, including limit cycle solutions such as in-phase oscillation and those involving phase shifts. We also show the existence of "submaximal" limit cycle solutions under generic conditions. The canonical invariant region of the phase space is defined and used to investigate the dynamics. We discuss how the limit cycles lose and gain stability, and how symmetry can give rise to structurally stable heteroclinic cycles, a phenomenon not generically found in systems without symmetry. We also investigate how certain types of coupling (including linear coupling between oscillators with symmetric waveforms) can give rise to degenerate behaviour, where the oscillators decouple into smaller groups.

Original languageEnglish (US)
Pages (from-to)69-108
Number of pages40
JournalJournal of Nonlinear Science
Volume2
Issue number1
DOIs
StatePublished - Mar 1992

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Phase shift
oscillators
Symmetry
symmetry
Limit Cycle
Invariant Region
cycles
Phase Shift
Phase Space
phase shift
Heteroclinic Cycle
permutations
Weak Coupling
Waveform
Permutation
waveforms
Classify
Oscillation
Ring
oscillations

Keywords

  • AMS/MOS classification numbers: 34C, 58F
  • bifurcation with symmetry
  • structurally stable heteroclinic cycles
  • weakly coupled oscillators

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Applied Mathematics
  • Mathematics(all)
  • Mechanics of Materials
  • Computational Mechanics

Cite this

The dynamics of n weakly coupled identical oscillators. / Ashwin, P.; Swift, James W.

In: Journal of Nonlinear Science, Vol. 2, No. 1, 03.1992, p. 69-108.

Research output: Contribution to journalArticle

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