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

Chaos. 2009 Dec;19(4):043104. doi: 10.1063/1.3247089.


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.

Publication types

  • Research Support, U.S. Gov't, Non-P.H.S.

MeSH terms

  • Algorithms*
  • Computer Simulation
  • Feedback
  • Models, Statistical*
  • Nonlinear Dynamics*
  • Oscillometry / methods*