# American Institute of Mathematical Sciences

• Previous Article
Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term
• DCDS-S Home
• This Issue
• Next Article
Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative
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
##### References:

show all references

##### 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 $v_1(t)$, $v_3(t)$, $v_4(t)$.
 [1] Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342 [2] Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341 [3] Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339 [4] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [5] Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318 [6] Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240 [7] Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048 [8] Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074 [9] Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103 [10] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468 [11] Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046 [12] Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336 [13] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [14] Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457 [15] Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464 [16] Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 [17] Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316 [18] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [19] Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219 [20] Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344

2019 Impact Factor: 1.233