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Journal of Nonlinear Science

, Volume 16, Issue 3, pp 201–231 | Cite as

Winding Numbers and Average Frequencies in Phase Oscillator Networks

  • M. Golubitsky
  • K. Josic
  • E. Shea-Brown
Article

Abstract

We study networks of coupled phase oscillators and show that network architecture can force relations between average frequencies of the oscillators. The main tool of our analysis is the coupled cell theory developed by Stewart, Golubitsky, Pivato, and Torok, which provides precise relations between network architecture and the corresponding class of ODEs in RM and gives conditions for the flow-invariance of certain polydiagonal subspaces for all coupled systems with a given network architecture. The theory generalizes the notion of fixed-point subspaces for subgroups of network symmetries and directly extends to networks of coupled phase oscillators. For systems of coupled phase oscillators (but not generally for ODEs in RM, where M ≥ 2), invariant polydiagonal subsets of codimension one arise naturally and strongly restrict the network dynamics. We say that two oscillators i and j coevolve if the polydiagonal θi = θj is flow-invariant, and show that the average frequencies of these oscillators must be equal. Given a network architecture, it is shown that coupled cell theory provides a direct way of testing how coevolving oscillators form collections with closely related dynamics. We give a generalization of these results to synchronous clusters of phase oscillators using quotient networks, and discuss implications for networks of spiking cells and those connected through buffers that implement coupling dynamics.

Keywords

Network Architecture Phase Equation Original Network Invariant Torus Phase Oscillator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer 2006

Authors and Affiliations

  • M. Golubitsky
    • 1
  • K. Josic
    • 1
  • E. Shea-Brown
    • 2
  1. 1.Department of Mathematics, University of Houston, Houston TX 77204-3008USA
  2. 2.Courant Institute of Mathematical Sciences and Center for Neural Science, 251 Mercer St., New York University, New York, NY 10012USA

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