Operator-theoretic Analysis of Mutual Interactions in Synchronized Dynamics

Abstract

Analyzing synchronized nonlinear oscillators is one of the most important and attractive topics in nonlinear science. By understanding the interactions between the oscillators, we can figure out the synchronization process. A promising approach to the analysis of interacting oscillators in nonlinear science is the application of the phase model. In this paper, we propose a data-driven approach to extract mutual interactions of synchronized oscillators based on the phase model. Recently, applying machine learning techniques to estimate models in physics has been actively investigated. We propose an operator-theoretic approach to estimate the phase model of interacting oscillators. We reduce the estimation problem to a multiparameter eigenvalue problem of the Koopman operator, a linear operator that describes a dynamical system. By reducing the problem to a linear algebraic problem, we can theoretically show that the proposed approach is stable with respect to perturbations in the given data.