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<p>Synchronization in coupled nonlinear oscillators is central to understanding collective behavior in physics, biology, and engineering. This work presents a Koopman operator-based framework for analyzing synchronization in diffusively coupled Van der Pol oscillators, with a brief extension to a bio-inspired fish robot model. Traditional time-domain and Hilbert-phase analyses identify phase coherence for sufficiently strong coupling, but the Koopman approach provides a richer operator-theoretic view by lifting nonlinear dynamics into a linear observable space. Extended Dynamic Mode Decomposition (EDMD) is employed to compute Koopman spectra and modes, revealing oscillatory stability and latent synchronization manifolds. Data-driven embeddings using Principal Component Analysis (PCA), t-distributed stochastic neighbor embedding (t-SNE), and K-means uncover coherent structures and phase-locking regimes. Further, residual diagnostics quantify model fidelity and allow estimation of coupling strength. Results demonstrate robust synchronization across small oscillator networks and confirm scalability up to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(N=100\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>100</mn> </mrow> </math></EquationSource> </InlineEquation>. The study highlights Koopman analysis as a powerful alternative to classical phase-reduction, offering both microscopic residual insights and macroscopic coherence characterization, with broader implications for nonlinear networks in engineering and bio-inspired systems.</p>

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Koopman-based synchronization analysis and residual diagnostics of coupled Van der Pol oscillators

  • 期刊论文

  • 发表日期：

    2026年5月13日

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  • N. P. Sasikumar,
  • P. Balasubramaniam

摘要

Synchronization in coupled nonlinear oscillators is central to understanding collective behavior in physics, biology, and engineering. This work presents a Koopman operator-based framework for analyzing synchronization in diffusively coupled Van der Pol oscillators, with a brief extension to a bio-inspired fish robot model. Traditional time-domain and Hilbert-phase analyses identify phase coherence for sufficiently strong coupling, but the Koopman approach provides a richer operator-theoretic view by lifting nonlinear dynamics into a linear observable space. Extended Dynamic Mode Decomposition (EDMD) is employed to compute Koopman spectra and modes, revealing oscillatory stability and latent synchronization manifolds. Data-driven embeddings using Principal Component Analysis (PCA), t-distributed stochastic neighbor embedding (t-SNE), and K-means uncover coherent structures and phase-locking regimes. Further, residual diagnostics quantify model fidelity and allow estimation of coupling strength. Results demonstrate robust synchronization across small oscillator networks and confirm scalability up to \(N=100\) N = 100 . The study highlights Koopman analysis as a powerful alternative to classical phase-reduction, offering both microscopic residual insights and macroscopic coherence characterization, with broader implications for nonlinear networks in engineering and bio-inspired systems.