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Spatially structured networks of pulse-coupled phase oscillators on metric spaces
Stilianos Louca and Fatihcan M. Atay
The Winfree model describes finite networks of phase oscillators that interact by broadcasting pulses modulating the current frequency of connected oscillators. We study a generalization of the model and its fluid-dynamical limit for networks, where oscillators are distributed on some abstract $\sigma$-finite Borel measure space over a separable metric space. We give existence and uniqueness statements for solutions to the continuity equation for the oscillator phase densities. We further show that synchrony in networks of identical oscillators is locally asymptotically stable for finite, strictly positive measures and under suitable conditions on the oscillator response function and the coupling kernel of the network. The conditions on the latter are a generalization of the strong connectivity of finite graphs to abstract convolution kernels.