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 one reaction–diffusion equation coupled with ordinary differential equation. This includes FitzHugh–Nagumo system, caricature calcium models, other models real-world applications. Based on recent results averaging theory, we solve solutions related three-dimensional systems then prove arising bifurcations. Numerical computation by Tsai et al. [SIAM J. Appl. Dyn. Syst. 11, 1149–1199 (2012)] once suggested that bifurcations model are spectrally unstable, yet without proof. After analyzing linearization about relatively bounded perturbation, instability through perturbation unstable spectra for equilibria. As an application, applied current.