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arXiv · 2504.06377

Geometric perspective of linear stability in finite networks of nonlinear oscillators

Yashee Sinha·Priya B. Jain·Antonio Mihara·Rene O. Medrano-T·Ján Mináč·Lyle E. Muller·Roberto C. Budzinski
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Abstract

We use a complex-valued transformation of the Kuramoto model to develop an operator-description of the linear stability in finite networks of nonlinear oscillators. This mathematical approach offers analytical predictions for the linear stability of $q$-states, which include phase synchronization ($q = 0$) and waves with different spatial frequencies ($|q| > 0$). This approach seamlessly incorporates the presence of time delays (represented by phase-lags in the coupling). With this, we are able to analytically determine the specific combination of connectivity and time delays (phase-lags) that leads to any given $q$-state to be linearly stable. This approach offers a geometric perspective of linear stability in finite networks in terms of the connectivity and delays (phase-lag), and it opens a path to designing and controlling the spatiotemporal dynamics of individual oscillator networks.

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Yashee Sinha, Priya B. Jain, Antonio Mihara, Rene O. Medrano-T, Ján Mináč, Lyle E. Muller, Roberto C. Budzinski. 2025-04-08. Geometric perspective of linear stability in finite networks of nonlinear oscillators. https://arxiv.org/abs/2504.06377

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