Solvable Dynamics of Coupled High-Dimensional Generalized Limit-Cycle Oscillators

TL;DR

This paper introduces a new high-dimensional coupled oscillator model that incorporates amplitude dynamics, generalizing the Kuramoto phase model and providing rigorous stability analysis and predictions for phenomena like amplitude death.

Contribution

The paper presents a novel $D$-dimensional oscillator model that explicitly includes amplitude dynamics, extending the Kuramoto model and offering rigorous stability and spectral analysis.

Findings

Incoherence is stable for negative coupling and unstable for positive coupling in 3D.

Locked states exist when coupling is positive.

Amplitude death onset is theoretically predicted.

Abstract

We introduce a new model consisting of globally coupled high-dimensional generalized limit-cycle oscillators, which explicitly incorporates the role of amplitude dynamics of individual units in the collective dynamics. In the limit of weak coupling, our model reduces to the $D$-dimensional Kuramoto phase model, akin to a similar classic construction of the well-known Kuramoto phase model from weakly coupled two-dimensional limit-cycle oscillators. For the practically important case of $D=3$, the incoherence of the model is rigorously proved to be stable for negative coupling $(K<0)$ but unstable for positive coupling $(K>0)$; the locked states are shown to exist if $K>0$; in particular, the onset of amplitude death is theoretically predicted. For $D\geq2$, the discrete and continuous spectra for both locked states and amplitude death are governed by two general formulas. Our proposed $D$-dimensional model is physically more reasonable, because it is no longer constrained by fixed amplitude dynamics, which puts the recent studies of the $D$-dimensional Kuramoto phase model on a stronger footing by providing a more general framework for $D$-dimensional limit-cycle oscillators.