Partially integrable dynamics of ensembles of nonidentical oscillators

AbstractWe consider ensembles of sine-coupled phase oscillators consisting of subpopulationsof identical units, with a general heterogeneous coupling between subpopulations.Using the Watanabe-Strogatz ansatz we reduce the dynamics of the ensemble to arelatively small numberof dynamical variables plusmicroscopic constants of motion.This reduction is independent of the sizes of subpopulations and remains valid inthe thermodynamic limits, where these sizes or/and the number of subpopulationsare infinite. We demonstrate that the approach to the dynamics of such systems,recently proposed by Ott and Antonsen, corresponds to a particular choice of micro-scopic constants of motion. The theory is applied to the standard Kuramoto modeland to the description of two interacting subpopulations, exhibiting a chimera state.Furthermore, we analyze the dynamics of the extension of the Kuramoto model forthe case of nonlinear coupling and demonstrate the multistability of synchronousstates. Keywords: Coupled oscillators, oscillator ensembles, Kuramoto model, nonlinearcoupling

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