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

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 Möbius 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.Large arrays of coupled limit-cycle oscillators have been used to model diverse systems in physics, biology, chemistry, engineering and social science. The special case of phase oscillators coupled all-to-all through sinusoidal interactions has attracted mathematical interest because of its analytical tractability. About 20 years ago, numerical experiments revealed that these systems display an exceptionally simple form of collective behavior: for all N ≥ 3, where N is the number of oscillators, all trajectories are confined to manifolds with N − 3 fewer dimensions than the state space itself. Several insights have been obtained over the past two decades, but it has remained an open problem to pinpoint the symmetry or other structure that causes this non-generic behavior. Here we show that group theory provides the explanation: the governing equations for these systems arise naturally from the action of the group of conformal mappings of the unit disc to itself. This link unifies and explains the previous numerical and analytical results, and yields new constants of motion for this class of dynamical systems.

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Захват Фазы Для Уравнений, Описывающих Резистивную Модель Джозефсоновского Перехода, И Их Возмущений

“…которое мы будем рассматривать не только при малом, но и при любом ε. Это уравнение используется для моделирования динамики перехода Джозефсона ( [7], [9], [11]), и его свойства изучаются как в этом контексте ( [2]- [4], [8], [13]), так и при решении других задач ( [10], [12]). Мы будем называть его уравнением класса Д. Физическая интерпретация этого уравнения обсуждается ниже.…”