Classical solutions of a mean field system for pulse-coupled oscillators: long time asymptotics versus blowup
arxiv(2024)
Abstract
We introduce a novel reformulation of the mean-field system for pulse-coupled
oscillators. It is based on writing a closed equation for the inverse
distribution function associated to the probability density of oscillators with
a given phase in a suitable time scale. This new framework allows to show a
hidden contraction/expansion of certain distances leading to a full
clarification of the long-time behavior, existence of steady states, rates of
convergence, and finite time blow-up of classical solutions for a large class
of monotone phase response functions. In the process, we get insights about the
origin of obstructions to global-in-time existence and uniform in time
estimates on the firing rate of the oscillators.
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