Two-cluster bifurcations in systems of globally pulse-coupled oscillators

TL;DR

This paper analyzes stability and bifurcations of synchronized and two-cluster states in globally pulse-coupled oscillators, revealing conditions for multistability and phenomena like intermittent synchronization.

Contribution

It derives stability conditions for all states in pulse-coupled oscillators using phase-response-curves and explores complex bifurcation structures and attractor formations.

Findings

Multiple stable two-cluster states can coexist in certain parameter regions.

Unstable one-cluster states can form attractors with homoclinic connections.

Intermittent synchronization occurs with diminishing beats away from perfect synchrony.

Abstract

For a system of globally pulse-coupled phase-oscillators, we derive conditions for stability of the completely synchronous state and all possible two-cluster states and explain how the different states are naturally connected via bifurcations. The coupling is modeled using the phase-response-curve (PRC), which measures the sensitivity of each oscillator's phase to perturbations. For large systems with a PRC, which turns to zero at the spiking threshold, we are able to find the parameter regions where multiple stable two-cluster states coexist and illustrate this by an example. In addition, we explain how a locally unstable one-cluster state may form an attractor together will its homoclinic connections. This leads to the phenomenon of intermittent, asymptotic synchronization with abating beats away from the perfect synchrony.