March  2010, 13(2): 503-516. doi: 10.3934/dcdsb.2010.13.503

Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay

1. 

Department of Mathematics, Computer & Information Sciences, Mississippi Valley State University, Itta Bena, MS 38941, United States

2. 

Department of Mathematics & Statistics, Kennesaw State University, Kennesaw, GA 30144

Received  February 2009 Revised  October 2009 Published  December 2009

In this paper, we study the following system of two coupled relaxation oscillators of the van der Pol type with delay

ε$\ddot{x}_1-(1-x_1^2)\dot{x}_1+x_1=h_1(x_2(t-\tau)-x_1(t-\tau)),$
ε$\ddot{x}_2-(1-x_2^2)\dot{x}_2+x_2=h_2(x_1(t-\tau)-x_2(t-\tau)),$

where $h_1$ and $h_2$ are nonlinear functions. It is shown that this system can exhibit Hopf bifurcation as the time delay $\tau$ passes certain critical values. The distribution of the eigenvalues of the linearized system is studied thoroughly in terms of the parameter $\ep$ and the linear parts of functions $h_1$ and $h_2$. The normal form theory for general retarded functional equations developed by Faria and Magalhães is applied to perform center manifold reduction and hence to obtain the explicit normal form Hopf bifurcation which can be used to determine the stability of the bifurcating periodic solutions and and the direction of Hopf bifurcation. Examples are given to confirm the theoretical results.

Citation: 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
[1]

Stefan Siegmund. Normal form of Duffing-van der Pol oscillator under nonautonomous parametric perturbations. Conference Publications, 2001, 2001 (Special) : 357-361. doi: 10.3934/proc.2001.2001.357

[2]

Zhaosheng Feng. Duffing-van der Pol-type oscillator systems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1231-1257. doi: 10.3934/dcdss.2014.7.1231

[3]

Zhaoxia Wang, Hebai Chen. A nonsmooth van der Pol-Duffing oscillator (I): The sum of indices of equilibria is $ -1 $. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021096

[4]

Zhaoxia Wang, Hebai Chen. A nonsmooth van der Pol-Duffing oscillator (II): The sum of indices of equilibria is $ 1 $. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021101

[5]

Zhaosheng Feng, Guangyue Gao, Jing Cui. Duffing--van der Pol--type oscillator system and its first integrals. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1377-1391. doi: 10.3934/cpaa.2011.10.1377

[6]

Kazuyuki Yagasaki. Existence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021151

[7]

Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643

[8]

Boris Anicet Guimfack, Conrad Bertrand Tabi, Alidou Mohamadou, Timoléon Crépin Kofané. Stochastic dynamics of the FitzHugh-Nagumo neuron model through a modified Van der Pol equation with fractional-order term and Gaussian white noise excitation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2229-2243. doi: 10.3934/dcdss.2020397

[9]

Qiaolin He, Chang Liu, Xiaoding Shi. Numerical study of phase transition in van der Waals fluid. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4519-4540. doi: 10.3934/dcdsb.2018174

[10]

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

[11]

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

[12]

Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529

[13]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997

[14]

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

[15]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805

[16]

Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363

[17]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109

[18]

Shu-Yi Zhang. Existence of multidimensional non-isothermal phase transitions in a steady van der Waals flow. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2221-2239. doi: 10.3934/dcds.2013.33.2221

[19]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084

[20]

Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]