Emergence of phase-locked states for a deterministic and stochastic Winfree model with inertia

TL;DR

This paper investigates how phase-locking and synchronization emerge in both deterministic and stochastic Winfree oscillator models with inertia, revealing new phenomena in small coupling regimes and under noise perturbations.

Contribution

It demonstrates the emergence of non-trivial synchronization in small coupling regimes and analyzes the effects of stochastic perturbations on Winfree oscillators with inertia.

Findings

Synchronization occurs in small coupling regimes.

Stochastic noise influences the emergence of phase-locking.

Numerical simulations suggest broader analytical possibilities.

Abstract

We study the emergence of phase-locking for Winfree oscillators under the effect of inertia. It is known that in a large coupling regime, oscillators governed by the deterministic second-order Winfree model with inertia converge to a unique equilibrium. In contrast, in this paper we show the asymptotic emergence of non-trivial synchronization in a suitably small coupling regime. Moreover, we study the effect of a new stochastically perturbed Winfree system with multiplicative noise and obtain lower estimates in probability for the pathwise emergence of such a synchronizing pattern, provided the noise is sufficiently small. We also provide numerical simulations which hint at the possibility of more general and stronger analytical results.