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

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, and other models in 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 by Tsai et al. [SIAM J. Appl. Dyn. Syst. 11, 1149-1199 (2012)] 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 linearization 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.