The mathematics of asymptotic stability in the Kuramoto model

Abstract

Now a standard in Nonlinear Sciences, the Kuramoto model is the perfect example of the transition to synchrony in heterogeneous systems of coupled oscillators. While its basic phenomenology has been sketched in early works, the corresponding rigorous validation has long remained problematic and was achieved only recently. This paper reviews the mathematical results on asymptotic stability of stationary solutions in the continuum limit of the Kuramoto model, and provides insights into the principal arguments of proofs. This review is complemented with additional original results, various examples, and possible extensions to some variations of the model in the literature.

20 pages

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• Exact Response Theory and Kuramoto dynamics
• On the trend to global equilibrium for Kuramoto Oscillators
• Orientation mixing in active suspensions
• Synchronization for Networks of Globally Coupled Maps in the Thermodynamic Limit
• Collective in-plane magnetization in a 2D XY macrospin system within the framework of generalized Ott-Antonsen theory
• Stability of partially locked states in the Kuramoto model through Landau damping with Sobolev regularity
• Computer-assisted proof of ergodicity breaking in expanding coupled maps
• A Neural Network for the Identical Kuramoto Equation: Architectural Considerations and Performance Evaluation