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

Shuang Chen, Jinqiao Duan

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.

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

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.

Paper Structure

This paper contains 14 sections, 15 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Existence of periodic traveling waves via averaging theory
  3. Fold-Hopf equilibrium
  4. Averaging theory
  5. Existence of periodic traveling waves
  6. Instability of periodic traveling waves via perturbation theory
  7. Spectral theory for periodic traveling waves
  8. Relatively bounded perturbation
  9. Instability of periodic traveling waves
  10. Applications to the FitzHugh-Nagumo system
  11. Mathematical model
  12. Fold-Hopf equilibrium
  13. Existence and stability of periodic traveling waves
  14. Concluding remarks

Key Result

Lemma \oldthetheorem

Assume that $P_{0}:=(u_{0},v_{0},w_{0})^{T}$ in $\mathbb{R}^{3}$ is a fold-Hopf equilibrium of 3D-SYSTEM with $(\alpha,c)=(\alpha_{0},c_{0})$ and $c_{0}\neq 0$. Let $\lambda_{1}=0$ and $\lambda_{2,3}=\pm\, {\bf i}\, \mu_{0}$ for some $\mu_{0}>0$ be the eigenvalues of the Jacobian matrix $J(u_{0},v_{

Theorems & Definitions (26)

  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • ...and 16 more