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arXiv · 0904.1680

Phase oscillators with global sinusoidal coupling evolve by Mobius group action

Seth A. Marvel·Renato E. Mirollo·Steven H. Strogatz
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Abstract

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 Mobius group, a three-parameter subgroup of fractional linear transformations that map the unit disc 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.

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Seth A. Marvel, Renato E. Mirollo, Steven H. Strogatz. 2009-07-25. Phase oscillators with global sinusoidal coupling evolve by Mobius group action. https://doi.org/10.1063/1.3247089

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