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

Runxia Wang; Haihong Liu; Fang Yan; Xiaohui Wang

doi:10.3934/dcdss.2017026

Article Contents

2017, Volume 10, Issue 3: 523-542. Doi: 10.3934/dcdss.2017026

This issue Previous Article Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative Next Article Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term

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
^* Corresponding author

Received: December 2015

Revised: November 2016

Published: May 2017

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 37G05, 37G15; Secondary: 34C23.

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Figure 1. The bifurcation diagrams for system (2) with the parameters $(\mu_1,\mu_2)$ around $(0,0)$.

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Figure 2. The phase portraits in $D_{1}-D_{6}$

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Figure 3. 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 $v_1(t)$, $v_2(t)$, $v_3(t)$.

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Figure 4. 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 $v_1(t)$, $v_3(t)$, $v_4(t)$.

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Figure 5. 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 $v_1(t)$, $v_3(t)$, $v_4(t)$.

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Figure 6. 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 $v_1(t)$, $v_3(t)$, $v_4(t)$.

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Figure 7. 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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