# 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] Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098 [2] P.E. Kloeden. Pitchfork and transcritical bifurcations in systems with homogeneous nonlinearities and an almost periodic time coefficient. Communications on Pure & Applied Analysis, 2004, 3 (2) : 161-173. doi: 10.3934/cpaa.2004.3.161 [3] Lin Zhao, Zhi-Cheng Wang, Liang Zhang. Threshold dynamics of a time periodic and two–group epidemic model with distributed delay. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1535-1563. doi: 10.3934/mbe.2017080 [4] Mikhail Kamenskii, Boris Mikhaylenko. Bifurcation of periodic solutions from a degenerated cycle in equations of neutral type with a small delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 437-452. doi: 10.3934/dcdsb.2013.18.437 [5] Shangbing Ai. Multiple positive periodic solutions for a delay host macroparasite model. Communications on Pure & Applied Analysis, 2004, 3 (2) : 175-182. doi: 10.3934/cpaa.2004.3.175 [6] Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325 [7] Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735 [8] Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121 [9] Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 [10] Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891 [11] Jisun Lim, Seongwon Lee, Yangjin Kim. Hopf bifurcation in a model of TGF-$\beta$ in regulation of the Th 17 phenotype. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3575-3602. doi: 10.3934/dcdsb.2016111 [12] Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657 [13] Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046 [14] Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182 [15] Xiaoqin P. Wu, Liancheng Wang. Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 503-516. doi: 10.3934/dcdsb.2010.13.503 [16] Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457 [17] Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031 [18] Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997 [19] Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 [20] John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805

2018 Impact Factor: 0.545

## Metrics

• HTML views (15)
• Cited by (0)

• on AIMS