Enlarged Kuramoto Model: Secondary Instability and Transition to Collective Chaos

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.

arXiv comment12 pages, 4 figures

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• Phase chimera states on non-local hyperrings
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• Analytical Phase Reduction for Weakly Nonlinear Oscillators
• Heteroclinic cycles and chaos in a system of four identical phase oscillators with global biharmonic coupling
• Efficient moment-based approach to the simulation of infinitely many heterogeneous phase oscillators