Stabilities and dynamic transitions of the Fitzhugh-Nagumo system

Chao Xing; Zhigang Pan; Quan Wang

doi:10.3934/dcdsb.2020134

Article Contents

2021, Volume 26, Issue 2: 775-794. Doi: 10.3934/dcdsb.2020134

This issue Previous Article Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control Next Article Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations

Stabilities and dynamic transitions of the Fitzhugh-Nagumo system

1.
College of Mathematics, Sichuan University, Chengdu, Sichuan, 610065, China
2.
School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, PR China
3.
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 610054, China

^* Corresponding author: Quan Wang
^* Corresponding author: Quan Wang

Received: January 2020

Early access: April 2020

Published: February 2021

This research was supported by the NSFC, Grant No.11901408 and 11711306

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Abstract

The article aims to examine the dynamic transition of the reaction-diffusion Fitzhugh-Nagumo system defined on a thin spherical shell and a 2D-rectangular domain. The mathematical tool employed is the theory of phase transition dynamics established for dissipative dynamical systems. The main results in this paper include two parts. First, for the system on a thin spherical shell, we only focus on the transition from a real simple eigenvalue. More precisely, if the first eigenspace is three–dimensional, the system undergoes either a continuous transition or a jump transition. Besides, a mix transition is also allowed if the first eigenspace is one–dimensional. Second, for the system on a rectangular domain, both the transitions from a simple real eigenvalue and a pair of simple complex eigenvalues are considered. Our results imply that two steady-state solutions bifurcate, which are either attractors or saddle points, and a Hopf bifurcation is also possible in the system on the rectangular domain.

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Mathematics Subject Classification: Primary: 35B32, 35B35; Secondary: 35Q92, 37L10.

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Figure 1. Neutral stability surface $ \tilde{\alpha}_{c}(D, \epsilon) $

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Figure 2. The regions separating two types of transitions. Region A, jump transitions from a simple real eigenvalue; Region B, jump transitions from a pair of simple complex eigenvalues

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