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

For a system of globally pulse-coupled phase-oscillators, we derive conditions for stability of the completely synchronous state and all stationary 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 is 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 with its homoclinic connections. This leads to the phenomenon of intermittent, asymptotic synchronization with abating beats away from the perfect synchrony.