Koopman analysis in oscillator synchronization

TL;DR

This paper extends Koopman operator analysis to coupled oscillator synchronization, enabling the identification of transition points and eigenfunctions from observed data, bridging local and global analysis methods.

Contribution

It introduces a Koopman-based framework for analyzing synchronization in coupled oscillators, including an analytic approximation and numerical validation for various coupling strengths.

Findings

Eigenvalues and eigenfunctions can be extracted from time series data.

Synchronization transition points are accurately identified via eigenfunction correlation.

The method applies to different coupling regimes, from weak to strong.

Abstract

Synchronization is an important dynamical phenomenon in coupled nonlinear systems, which has been studied extensively in recent years. However, analysis focused on individual orbits seems hard to extend to complex systems while a global statistical approach is overly cursory. Koopman operator technique seems to well balance the two approaches. In this paper, we extend Koopman analysis to the study of synchronization of coupled oscillators by extracting important eigenvalues and eigenfunctions from the observed time series. A renormalization group analysis is designed to derive an analytic approximation of the eigenfunction in case of weak coupling that dominates the oscillation. For moderate or strong couplings, numerical computation further confirms the importance of the average frequencies and the associated eigenfunctions. The synchronization transition points could be located with quite high accuracy by checking the correlation of neighbouring eigenfunctions at different coupling strengths, which is readily applied to other nonlinear systems.