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.

12 pages, 4 figures

References in corpus (7)

• Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators
• Three-body forces: From cold atoms to nuclei
• Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplices
• Coupling functions: Universal insights into dynamical interaction mechanisms
• Phase reduction beyond the first order: the case of the mean-field complex Ginzburg-Landau equation
• On the Concept of Dynamical Reduction : The Case of Coupled Oscillators
• Finite-size-induced transitions to synchrony in oscillator ensembles with nonlinear global coupling

Cited by in corpus (2)

• Solvable Dynamics of Coupled High-Dimensional Generalized Limit-Cycle Oscillators
• Analytical Phase Reduction for Weakly Nonlinear Oscillators