Instability of small-amplitude periodic waves from fold-Hopf bifurcation

TL;DR

This paper investigates the existence and spectral instability of small-amplitude periodic waves emerging from fold-Hopf bifurcations in reaction-diffusion systems, including the FitzHugh-Nagumo model, using averaging theory and spectral analysis.

Contribution

It provides a rigorous proof of the instability of small-amplitude periodic waves from fold-Hopf bifurcations, extending previous numerical observations.

Findings

Small-amplitude periodic waves are spectrally unstable.

Existence of periodic waves from fold-Hopf bifurcations is established.

Application to FitzHugh-Nagumo system confirms theoretical results.

Abstract

We study the existence and stability of small-amplitude periodic waves emerging from fold-Hopf equilibria in a system of one reaction-diffusion equation coupled with one ordinary differential equation. This coupled system includes the FitzHugh-Nagumo system, caricature calcium models, consumer-resource models and other models in the real-world applications. Based on the recent results on the averaging theory, we solve periodic solutions in related three-dimensional systems and then prove the existence of periodic waves arising from fold-Hopf bifurcations. Numerical computation in [J. Tsai, W. Zhang, V. Kirk, and J. Sneyd, SIAM J. Appl. Dyn. Syst. 11 (2012), 1149--1199] once suggested that the periodic waves from fold-Hopf bifurcations in a caricature calcium model are spectrally unstable, yet without a proof. After analyzing the linearization about periodic waves by the relatively bounded perturbation, we prove the instability of small-amplitude periodic waves through a perturbation of the unstable spectra for the linearizations about the fold-Hopf equilibria. As an application, we prove the existence and stability of small-amplitude periodic waves from fold-Hopf bifurcations in the FitzHugh-Nagumo system with an applied current.