Almost global synchronization of amplitude-dependent high-dimensional Kuramoto model

Abstract

The high-dimensional Kuramoto model (HDKM) on the unit sphere is commonly used to explain the phase synchronization of coupled oscillators in dynamic systems. However, the current model featuring fixed-amplitude oscillators cannot characterize some systems with varying-amplitude oscillators, such as optical arrays, satellite clusters. Herein, an amplitude-dependent HDKM (AHDKM), defined in a linear space rather than on a unit sphere, is first proposed. This model incorporating amplitude dynamics can be reduced to the HDKM for any coupling strength among oscillators. Next, oscillator distributions at equilibrium points are accurately described to facilitate the analysis of the AHDKM convergence. To determine the global attractivity of equilibrium point set, an easily verifiable sufficient criterion is established by a height function constructed at equilibrium points instead of a strict “Lyapunov function”. Based on this criterion, almost global synchronization of the AHDKM is rigorously proved under different connected graphs via the derived instability of non-synchronized equilibrium points. Finally, main theoretical results are verified through numerical simulations.