Emergent behaviors of Winfree oscillators on special orthogonal group

Abstract

We propose a generalized matrix-valued synchronization model which can be regarded as matrix generalization of the classical Winfree model to the special orthogonal group, and we provide several sufficient frameworks leading to the emergent behaviors of the Winfree matrix model. For $SO(2)$ case, the proposed model reduces to the classical Winfree model. For the general (non-identical) case, we prove the existence of a positively invariant trapping region, establish a leader--follower mechanism in which sufficiently strong coupling draws all oscillators into a neighborhood of the identity whenever at least one oscillator is initially nearby, and show $\ell^1$-exponential stability of solutions, from which we deduce existence, uniqueness, and exponential convergence to an equilibrium. In the identical-oscillator regime, we show that complete state synchronization and oscillator death both occur exponentially fast with an explicit decay rate, and we classify all equilibrium configurations as solutions to a fixed-point equation for the mean influence.