Collective Dynamics in Infinite Networks of Pulse-Coupled Phase Oscillators

This talk will present our work on networks of pulse-coupled phase oscillators in the thermodynamic limit, that is, with the network size tending to infinity. Two models are considered in which oscillators are distributed on a separable metric space, according to a finite Borel measure. The first model is an evolution equation for the oscillator phase field, the second model is a continuity equation in the oscillator phase distribution. Both models are examined with respect to the existence and local stability of synchrony, i.e. all oscillators having one common, time-dependent phase. Continuing, all-to-all pulse-coupled networks of phase oscillators with additive white noise are examined, in the limit where pulses tend to Dirac distributions. This is done using a Fokker-Planck equation for the oscillator phase density. Particular interest is devoted to stationary states (i.e. time-independent phase densities), their existence, uniqueness, stability and bifurcation behaviour at changing noise strength.