# American Institute of Mathematical Sciences

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June  2017, 10(3): 523-542. doi: 10.3934/dcdss.2017026

## Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay

 1 Department of mathematics, Yunnan Normal University, Kunming, Yunnan 650500, China 2 School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, TX 78539, USA

* Corresponding author

Received  December 2015 Revised  November 2016 Published  February 2017

Fund Project: The authors are supported by NNSFC (11562021/11572278/11526182) and the Science Foundations (2014FB138/2015FB140/YJG2014-B07) of Yunnan Province.

In this paper, we study a coupled FitzHugh-Nagumo (FHN) neurons model with time delay. The existence conditions on Hopf-pitchfork singularity are given. By selecting the coupling strength and time delay as the bifurcation parameters, and by means of the center manifold reduction and normal form theory, the normal form for this singularity is found to analyze the behaviors of the system. We perform the bifurcation analysis and numerical simulations, and present the bifurcation diagrams. Some interesting phenomena are observed, such as the existence of a stable fixed point, a stable periodic solution, a pair of stable fixed points, and the coexistence of a pair of stable fixed points and a stable periodic solution near the Hopf-pitchfork critical point.

Citation: Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026
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##### References:
The bifurcation diagrams for system (2) with the parameters $(\mu_1,\mu_2)$ around $(0,0)$.
The phase portraits in $D_{1}-D_{6}$
A stable trivial equilibria: $(a)$ waveform diagram for variable $v_1(t)$, $(b)$ waveform diagram for variable $v_2(t)$, $(c)$ waveform diagram for variable $v_3(t)$, $(d)$ waveform diagram for variable $v_4(t)$, and $(e)$ phase diagram for variable $v_1(t)$, $v_2(t)$, $v_3(t)$.
A stable periodic solution: $(a)$ waveform diagram for variable $v_1(t)$, $(b)$ waveform diagram for variable $v_2(t)$, $(c)$ waveform diagram for variable $v_3(t)$, $(d)$ waveform diagram for variable $v_4(t)$, $(e)$ phase diagram for variable $v_1(t)$, $v_2(t)$, $v_3(t)$, and $(f)$ phase diagram for variable $v_1(t)$, $v_3(t)$, $v_4(t)$.
A stable periodic solution: $(a)$ waveform diagram for variable $v_1(t)$, $(b)$ waveform diagram for variable $v_2(t)$, $(c)$ waveform diagram for variable $v_3(t)$, $(d)$ waveform diagram for variable $v_4(t)$, $(e)$ phase diagram for variable $v_1(t)$, $v_2(t)$, $v_3(t)$, and $(f)$ phase diagram for variable $v_1(t)$, $v_3(t)$, $v_4(t)$.
A pair of stable fixed points and a stable periodic solution coexist: $(a)$ waveform diagram for variable $v_1(t)$, $(b)$ waveform diagram for variable $v_2(t)$, $(c)$ waveform diagram for variable $v_3(t)$, $(d)$ waveform diagram for variable $v_4(t)$, $(e)$ phase diagram for variable $v_1(t)$, $v_2(t)$, $v_3(t)$, and $(f)$ phase diagram for variable $v_1(t)$, $v_3(t)$, $v_4(t)$.
A pair of stable fixed points: $(a)$ waveform diagram for variable $v_1(t)$, $(b)$ waveform diagram for variable $v_2(t)$, $(c)$ waveform diagram for variable $v_3(t)$, $(d)$ waveform diagram for variable $v_4(t)$, $(e)$ phase diagram for variable $v_1(t)$, $v_2(t)$, $v_3(t)$, and $(f)$ phase diagram for variable $v_1(t)$, $v_3(t)$, $v_4(t)$.
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