Generalization of the Winfree model to the high-dimensional sphere and its emergent dynamics

TL;DR

This paper extends the classical Winfree model of synchronization from the circle to high-dimensional spheres, introducing new conditions for phase-locking, stability, and equilibrium states in high-dimensional oscillator systems.

Contribution

The paper generalizes the Winfree model to high-dimensional spheres and establishes new conditions for phase-locking, stability, and equilibrium states in this extended framework.

Findings

New conditions for complete phase-locking states in the high-dimensional sphere model

Proof of exponential $ ext{l}^1$-stability of the equilibrium solutions

Existence of the oscillator death state in the high-dimensional Winfree sphere model

Abstract

We present a high-dimensional Winfree model in this paper. The Winfree model is the first mathematical model for synchronization on the circle. We generalize this model to the high-dimensional sphere and we call it "the Winfree sphere model." We restricted the support of the influence function in the neighborhood of the attraction point with a small diameter to mimic the influence function as the Dirac-delta distribution. Restricting the support of the influence function allows several new conditions of the complete phase-locking states for the identical Winfree sphere model compare to previous results. We also provide the exponential $\ell^1$-stability and the existence of the equilibrium solution to obtain the complete oscillator death state of the Winfree sphere model.