Identical phase oscillators with global sinusoidal coupling evolve by Möbius group action
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- Seth A. Marvel
- Cornell University 1 Center for Applied Mathematics, , Ithaca, New York 14853, USA
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- Renato E. Mirollo
- Boston College 2 Department of Mathematics, , Chestnut Hill, Massachusetts 02167, USA
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- Steven H. Strogatz
- Cornell University 1 Center for Applied Mathematics, , Ithaca, New York 14853, USA
書誌事項
- 公開日
- 2009-10-15
- DOI
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- 10.1063/1.3247089
- 公開者
- AIP Publishing
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説明
<jats:p>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.</jats:p>
収録刊行物
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- Chaos: An Interdisciplinary Journal of Nonlinear Science
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Chaos: An Interdisciplinary Journal of Nonlinear Science 19 (4), 043104-, 2009-10-15
AIP Publishing