A global bifurcation organizing rhythmic activity in a coupled network

Abstract

We study a system of coupled phase oscillators near a saddle-node on an invariant circle bifurcation and driven by random intrinsic frequencies. Under the variation of control parameters, the system undergoes a phase transition changing the qualitative properties of collective dynamics. Using the Ott-Antonsen reduction and geometric techniques for ordinary differential equations, we identify a heteroclinic bifurcation in a family of vector fields on a cylinder, which explains the change in collective dynamics. Specifically, we show that the heteroclinic bifurcation separates two topologically distinct families of limit cycles: contractible limit cycles before the bifurcation from noncontractibile ones after the bifurcation. Both families are stable for the model at hand.

arXiv comment22 pages, 12 figures

References in corpus (17)

• Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators
• Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators
• Coexistence of Coherence and Incoherence in Nonlocally Coupled Phase Oscillators
• Long Time Evolution of Phase Oscillator Systems
• Coexistence of synchrony and incoherence in oscillatory media under nonlinear global coupling
• Stability diagram for the forced Kuramoto model
• Global firing induced by noise or diversity in excitable media
• A proof of the Kuramoto conjecture for a bifurcation structure of the infinite dimensional Kuramoto model
• Non-universal results induced by diversity distribution in coupled excitable systems
• Stability and bifurcation for the Kuramoto model
• Bifurcations in the Kuramoto model on graphs
• Asymptotic description of transients and synchronized states of globally coupled oscillators
• Two Scenarios for the Onset and Suppression of Collective Oscillations in Heterogeneous Populations of Active Rotators
• The geometry of spontaneous spiking in neuronal networks
• Mean-field dynamics of a random neural network with noise
• Birth and destruction of collective oscillations in a network of two populations of coupled type 1 neurons
• Bifurcations of heteroclinic contours in two-parameter planar systems: Overview and explicit examples