Bifurcations in the Kuramoto model on graphs

Abstract

In his classical work, Kuramoto analytically described the onset of synchronization in all-to-all coupled networks of phase oscillators with random intrinsic frequencies. Specifically, he identified a critical value of the coupling strength, at which the incoherent state loses stability and a gradual build-up of coherence begins. Recently, Kuramoto's scenario was shown to hold for a large class of coupled systems on convergent families of deterministic and random graphs. Guided by these results, in the present work, we study several model problems illustrating the link between network topology and synchronization in coupled dynamical systems. First, we identify several families of graphs, for which the transition to synchronization in the Kuramoto model starts at the same critical value of the coupling strength and proceeds in practically the same way. These examples include Erdős-Rényi random graphs, Paley graphs, complete bipartite graphs, and certain stochastic block graphs. These examples illustrate that some rather simple structural properties such as the volume of the graph may determine the onset of synchronization, while finer structural features may affect only higher order statistics of the transition to synchronization. Further, we study the transition to synchronization in the Kuramoto model on power law and small-world random graphs. The former family of graphs endows the Kuramoto model with very good synchronizability: the synchronization threshold can be made arbitrarily low by varying the parameter of the power law degree distribution. For the Kuramoto model on small-world graphs, in addition to the transition to synchronization, we identify a new bifurcation leading to stable random twisted states. The examples analyzed in this work complement the results in [Chiba, Medvedev, The mean field analysis for the Kuramoto model on graphs (parts I and II), arxiv].

arXiv comment18 pages, 12 figures

References in corpus (7)

• The nonlinear heat equation on W-random graphs
• A proof of the Kuramoto conjecture for a bifurcation structure of the infinite dimensional Kuramoto model
• Stability and bifurcation for the Kuramoto model
• Small-world networks of Kuramoto oscillators
• The semilinear heat equation on sparse random graphs
• Stability of twisted states in the Kuramoto model on Cayley and random graphs
• Stability of twisted states in the continuum Kuramoto model

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• Phase Oscillator Networks with Nonlocal Higher-Order Interactions: Twisted States, Stability and Bifurcations
• Continuum limits of coupled oscillator networks depending on multiple sparse graphs
• Pattern Formation in Random Networks Using Graphons
• Mean-field limit of non-exchangeable systems
• Chimeras unfolded
• Dynamics of oscillator populations with disorder in the coupling phase shifts
• Bifurcations and patterns in the Kuramoto model with inertia
• Graphop Mean-Field Limits and Synchronization for the Stochastic Kuramoto Model
• Critical exponents in coupled phase-oscillator models on small-world networks
• Stability of twisted states on lattices of Kuramoto oscillators
• A global bifurcation organizing rhythmic activity in a coupled network
• Dynamics of large oscillator populations with random interactions
• Partially phase-locked solutions to the Kuramoto model
• Metastability in the stochastic nearest-neighbor Kuramoto model of coupled phase oscillators
• Chimeras with uniformly distributed heterogeneity: two coupled populations
• Synchronization modes in bipartite oscillator networks