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

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.