Enlarged Kuramoto Model: Secondary Instability and Transition to Collective Chaos

TL;DR

This paper extends the Kuramoto model to include quadratic nonlinearities, revealing complex phenomena like secondary instabilities and transitions to collective chaos in coupled oscillators.

Contribution

It introduces an enlarged Kuramoto model with three-body interactions, capturing nonlinear effects neglected in the original model.

Findings

Secondary instability destabilizes synchronized states.

Bifurcations lead to collective chaos.

Fourier-Hermite method enables efficient numerical analysis.

Abstract

The emergence of collective synchrony from an incoherent state is a phenomenon essentially described by the Kuramoto model. This canonical model was derived perturbatively, by applying phase reduction to an ensemble of heterogeneous, globally coupled Stuart-Landau oscillators. This derivation neglects nonlinearities in the coupling constant. We show here that a comprehensive analysis requires extending the Kuramoto model up to quadratic order. This "enlarged Kuramoto model" comprises three-body (nonpairwise) interactions, which induce strikingly complex phenomenology at certain parameter values. As the coupling is increased, a secondary instability renders the synchronized state unstable, and subsequent bifurcations lead to collective chaos. An efficient numerical study of the thermodynamic limit, valid for Gaussian heterogeneity, is carried out by means of a Fourier-Hermite decomposition of the oscillator density.