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Identical phase oscillators with global sinusoidal coupling evolve by Möbius group action

Chaos 19, 043104 (2009); doi:10.1063/1.3247089

Published 15 October 2009

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Seth A. Marvel,1 Renato E. Mirollo,2 and Steven H. Strogatz1
1Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA
2Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02167, USA
Systems of N identical phase oscillators with global sinusoidal coupling are known to display low-dimensional dynamics. Although this phenomenon was first observed about 20 years ago, its underlying cause has remained a puzzle. Here we expose the structure working behind the scenes of these systems by proving that the governing equations are generated by the action of the Möbius group, a three-parameter subgroup of fractional linear transformations that map the unit disk to itself. When there are no auxiliary state variables, the group action partitions the N-dimensional state space into three-dimensional invariant manifolds (the group orbits). The N−3 constants of motion associated with this foliation are the N−3 functionally independent cross ratios of the oscillator phases. No further reduction is possible, in general; numerical experiments on models of Josephson junction arrays suggest that the invariant manifolds often contain three-dimensional regions of neutrally stable chaos. ©2009 American Institute of Physics
History: Received 23 July 2009; accepted 21 September 2009; published 15 October 2009
Permalink: http://link.aip.org/link/?CHAOEH/19/043104/1
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KEYWORDS and PACS

Keywords
PACS
  • 05.45.Xt
    Synchronization; coupled oscillators (nonlinear dynamical systems)
  • 02.20.-a
    Group theory
  • YEAR: 2009

PUBLICATION DATA

ISSN:
1054-1500 (print)   1089-7682 (online)
Publisher:
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