Global Phase Synchronization Decoupled from Amplitude Dynamics

TL;DR

This paper introduces a novel method using Koopman operator theory to construct coupled-oscillator models that achieve global phase synchronization while allowing complex amplitude dynamics, extending the applicability of Kuramoto models.

Contribution

The authors develop a framework to preserve Kuramoto-type phase dynamics globally in coupled oscillators, even with complex amplitude behaviors, using phase-amplitude coordinates.

Findings

Introduced the Kuramoto-Stuart-Landau model with nontrivial synchronized dynamics

Constructed oscillators with Lorenz-type chaotic amplitude dynamics

Achieved global phase synchronization in general limit-cycle systems

Abstract

The Kuramoto model is a canonical framework for analyzing phase synchronization, yet its utility is restricted to the vicinity of the oscillator's unperturbed limit cycle. Here, we present a method to construct coupled-oscillator models that globally preserve Kuramoto-type phase dynamics by using phase-amplitude coordinates defined via Koopman operator theory. We introduce a solvable model, termed Kuramoto-Stuart-Landau model, which exhibits nontrivial synchronized dynamics far from the limit cycle. We also construct three phase synchronized oscillators whose amplitudes exhibit Lorenz-type chaos. Our method is applicable to general limit-cycle systems, achieving phase synchronization globally while preserving arbitrarily complex amplitude dynamics.