# American Institute of Mathematical Sciences

## 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

Received  January 2020 Published  April 2020

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

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.

Citation: Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020134
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Neutral stability surface $\tilde{\alpha}_{c}(D, \epsilon)$
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