Communications in Mathematical Sciences

Volume 21 (2023)

Number 7

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

Pages: 1875 – 1894



Myeongju Kang (School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea)

Marco Rehmeier (Faculty of Mathematics, Bielefeld University, Bielefeld, Germany)


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.


Winfree model, inertia, multiplicative noise, synchronization

2010 Mathematics Subject Classification

34F05, 70F40, 92B25

Received 22 July 2022

Received revised 22 December 2022

Accepted 20 January 2023

Published 9 October 2023