Enlarged Kuramoto model: Secondary instability and transition to collective chaos

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.

Automated summary

The emergence of collective synchrony from an incoherent state is described by the Kuramoto model, which needs to be extended to quadratic order for comprehensive analysis. The extended model exhibits complex phenomena, such as secondary instability and collective chaos, at certain parameter values.