Identical phase oscillators with global sinusoidal coupling evolve by Mobius group action.
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 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.
Marvel
Seth A
SA
Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA. sam255@cornell.edu
Mirollo
Renato E
RE
Strogatz
Steven H
SH
D000465:Algorithms; D003198:Computer Simulation; D005246:Feedback; D015233:Models, Statistical; D017711:Nonlinear Dynamics; D009991:Oscillometry / Q000379:methods*