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November  2018, 23(9): 3935-3947. doi: 10.3934/dcdsb.2018118

## Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain

 Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

Received  May 2017 Revised  October 2017 Published  April 2018

Fund Project: The author is grateful for Professor Shouhong Wang for his advice and suggestions.This research is supported in part by the National Science Foundation (NSF) grant DMS-1515024, and by the Office of Naval Research (ONR) grant N00014-15-1-2662.

The main objective of this article is to study the dynamic transitions of the FitzHugh-Nagumo equations on a finite domain with the Neumann boundary conditions and with uniformly injected current. We show that when certain parameter conditions are satisfied, the system undergoes a continuous dynamic transition to a limit cycle. A mixed type transition is also found when other conditions are imposed on the parameters. The main method used here is Ma & Wang's dynamic transition theory, which can be used generally on different set-ups for the FitzHugh-Nagumo equations.

Citation: Yiqiu Mao. Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3935-3947. doi: 10.3934/dcdsb.2018118
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