Spatially structured networks of pulse-coupled phase oscillators on metric spaces
This paper develops a very general framework for studying networks of coupled phase oscillators of the Winfree type. Here the oscillators are distributed over an arbitrary separable metric space \(X\), with a \(\sigma\)-finite Borel measure describing the distribution of oscillators in different region of the space, and the interaction between two oscillators depending on their positions in the space \(X\) through a coupling kernel. Two models are considered, one (the field model) in which there is a unique phase at each point of \(X\), and one (the fluid model) in which there is a probability distribution of phases at each point, evolving according to a continuity equation. Relations between the two models are discussed. Existence and uniqueness of solutions for the dynamical equations is proved in a suitable functional-analytic framework. In the case of identical oscillators, conditions for local asymptotic stability of synchronized oscillations are studied, for both the field and the fluid models. These involve the notion of strongly connecting coupling kernel, which generalizes the concept of strong connectivity of finite graphs, as well as conditions on the pulse and response functions of the oscillators.
scientific article; zbMATH DE number 6375852
scientific article