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Operator-theoretic Analysis of Mutual Interactions in Synchronized Dynamics

Yuka Hashimoto, Masahiro Ikeda, Hiroya Nakao, Yoshinobu Kawahara

TL;DR

The paper tackles understanding mutual interactions in synchronized nonlinear oscillators by introducing KGME, a data-driven operator-theoretic framework based on the Koopman operator to estimate phase dynamics from time-series data. By recasting the phase-interaction estimation as a generalized multiparameter eigenvalue problem and leveraging multiple Koopman eigenvectors, KGME robustly recovers the phase function and a Fourier-based phase coupling function with guaranteed stability under data perturbations. The authors provide theoretical stability results showing the learned parameters are insensitive to the number of Fourier components M and harmonics j, and validate the approach on the van der Pol and FitzHugh-Nagumo models as well as real circadian data from cohabiting mice. The method advances robust, data-driven synchronization analysis and strengthens the connection between operator theory and nonlinear dynamics for practical applications in biology and engineering.

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.

Operator-theoretic Analysis of Mutual Interactions in Synchronized Dynamics

TL;DR

The paper tackles understanding mutual interactions in synchronized nonlinear oscillators by introducing KGME, a data-driven operator-theoretic framework based on the Koopman operator to estimate phase dynamics from time-series data. By recasting the phase-interaction estimation as a generalized multiparameter eigenvalue problem and leveraging multiple Koopman eigenvectors, KGME robustly recovers the phase function and a Fourier-based phase coupling function with guaranteed stability under data perturbations. The authors provide theoretical stability results showing the learned parameters are insensitive to the number of Fourier components M and harmonics j, and validate the approach on the van der Pol and FitzHugh-Nagumo models as well as real circadian data from cohabiting mice. The method advances robust, data-driven synchronization analysis and strengthens the connection between operator theory and nonlinear dynamics for practical applications in biology and engineering.

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.
Paper Structure (25 sections, 6 theorems, 60 equations, 4 figures, 3 tables)

This paper contains 25 sections, 6 theorems, 60 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Phase reduction of weakly interacting oscillators
  4. Koopman operator
  5. KGME approach for estimating phase models
  6. Estimation of the phase function
  7. Estimation of the phase coupling function
  8. Practical computations
  9. Problem formulation
  10. Optimization
  11. Stability analysis of the KGME approach
  12. Comparison to the approach with computing the power of the eigenvector
  13. Comparison to the approach with the Fourier series
  14. Numerical results
  15. van der Pol model
  16. ...and 10 more sections

Key Result

Lemma 1

Let $u\in\tilde{\mathcal{H}}$ be the eigenvector of $\tilde{K}$ corresponding to the eigenvalue $\mathrm{e}^{\sqrt{-1}\omega \Delta t}$. Then, if $u^j\in\tilde{\mathcal{H}}$, then it is the eigenvector corresponding to the eigenvalue $\mathrm{e}^{\sqrt{-1}j\omega \Delta t}$ for $j\in\mathbb{Z}$. Her

Figures (4)

  • Figure 1: Phase reduction of coupled nonlinear oscillators to the phase model.
  • Figure 2: Sensitivity of the gradient. (The average value of five different $S'$.)
  • Figure 3: The function $\Gamma_d$ computed from the estimated phase coupling function. The blue line is computed by the exact model using the Fourier averaging approach mauroy12
  • Figure 4: The time-series data and the function $\Gamma_d$ computed from the estimated phase coupling function.

Theorems & Definitions (9)

  • Lemma 1
  • Remark 2
  • Theorem 3
  • Remark 4
  • Theorem 5
  • Lemma 6
  • Proposition 7
  • Proposition 8
  • Remark 9