Nonlinear Stability of Large-Period Traveling Waves Bifurcating from the Heteroclinic Loop in the FitzHugh-Nagumo Equation

TL;DR

This work analyzes the nonlinear stability of large-period traveling waves bifurcating from a heteroclinic loop in the FitzHugh-Nagumo equation, a reaction-diffusion system with degenerate diffusion. It develops a general spectral framework using Lyapunov-Schmidt reduction and the Lin-Sandstede method to describe small-modulus spectra around the periodic waves, and applies Bloch-wave techniques to obtain diffusive spectral stability with a simple zero eigenvalue. The authors then prove nonlinear stability by introducing a spatiotemporal phase modulation and coupling it with modulated and unmodulated perturbations, leveraging semigroup decomposition to overcome slow decay and deriving sharp decay rates. A detailed examination of the FHN example yields explicit spectral expansions near the origin, validates diffusion-like decay, and establishes a robust nonlinear damping mechanism that ensures global stability for small localized perturbations. Overall, the results provide a rigorous framework for stability of heteroclinic-bifurcating, large-period patterns in degenerate RD systems, with potential implications for pattern formation in excitable media.

Abstract

A wave front and a wave back that spontaneously connect two hyperbolic equilibria, known as a heteroclinic wave loop, give rise to periodic waves with arbitrarily large spatial periods through the heteroclinic bifurcation. The nonlinear stability of these periodic waves is established in the setting of the FitzHugh-Nagumo equation, which is a well-known reaction-diffusion model with degenerate diffusion. First, for general systems, we give the expressions of spectra with small modulus for linearized operators about these periodic waves via the Lyapunov-Schmidt reduction and the Lin-Sandstede method. Second, applying these spectral results to the FitzHugh-Nagumo equation, we establish their diffusive spectral stability. Finally, we consider the nonlinear stability of these periodic waves against localized perturbations. We introduce a spatiotemporal phase modulation $\varphi$, and couple it with the associated modulated perturbation $\mathbf{V}$ along with the unmodulated perturbation $\mathbf{\widetilde{V}}$ to close a nonlinear iteration argument.

Figures (3)

• Figure 1: $e_1$ and $e_2$ are equilibrium points of system \ref{['201']}. The black solid line from $e_1$ to $e_2$ and the black solid line from $e_2$ to $e_1$ represent the heteroclinic orbits $h_1(x)$ and $h_2(x)$, respectively. The red solid line represents the periodic orbit $p(x)$ bifurcating from the heteroclinic loop. In the schematic diagram, we divide the periodic orbit into four segments at $x=-L_1$( or equivalently $L_1+2L_2$), $0$, $L_1$, and $L_1+L_2$.
• Figure 2: (a) System (\ref{['403']}) has three equilibrium points. $\gamma =\gamma_0$ is chosen so that $e_1$ and $e_2$ are symmetric with respect to the inflection point $p(\bar{u},0,\bar{w})$ of the cubic function $w=u(1-u)(u-a)$. (b) Heteroclinic solutions $h_1(x)$ and $h_2(x)$ exist at the same time for a certain $(\gamma,c) = (\gamma_0, c(\epsilon))$, $h_1(x)$ from $e_1$ to $e_2$, and $h_2(x)$ from $e_2$ to $e_1$ at the same time, forming a heteroclinic loop.
• Figure 3: A bifurcation diagram of the equation (\ref{['403']}) with respect to the parameters $(\gamma, c)$. The parameter curves $\gamma_1(c)$ and $\gamma_2(c)$ represent the existence of homoclinic orbits to the equilibrium points $e_1$ and $e_2$, respectively. The shaded area between the curves $\gamma_1(c)$ and $\gamma_2(c)$ corresponds to the parameter region where periodic orbits exist.

Theorems & Definitions (55)

• Definition 1.1
• Theorem 1.2
• Lemma 2.1
• Theorem 2.2
• Remark 2.3
• proof
• Theorem 2.4
• proof
• Lemma 2.5
• proof