Periodic solutions for van der Pol equation with time delay

For van der Pol equation with distributed delay, when the distributed delay kernel is the general Gamma distributed delay kernel, the existence and the stability of periodic solution are obtained by using the linear chain trick and geometric singular perturbation theory. On the other hand, for van der Pol equation with discrete delay, the existence and the stability are proved by employing a technique which is based on inertial manifold. Finally, numerical simulations are also given to illustrate the results.

[1]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[2]  Christopher Jones,et al.  Geometric singular perturbation theory , 1995 .

[3]  Fatihcan M. Atay,et al.  VAN DER POL'S OSCILLATOR UNDER DELAYED FEEDBACK , 1998 .

[4]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[5]  Carmen Chicone,et al.  Inertial Flows, Slow Flows, and Combinatorial Identities for Delay Equations , 2004 .

[6]  G. M. Abd El-Latif.  Parametric excitation for forcing van der Pol oscillator , 2004, Appl. Math. Comput..

[7]  Kwok-Wo Wong,et al.  Stability of bifurcating periodic solutions for van der Pol equation with continuous distributed delay , 2003, Appl. Math. Comput..

[8]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

[9]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[10]  Jinde Cao,et al.  Hopf bifurcation and stability of periodic solutions for van der Pol equation with time delay , 2005 .

[11]  Shigui Ruan,et al.  Convergence and Travelling Fronts in Functional Differential Equations with Nonlocal Terms: A Competition Model , 2003, SIAM J. Math. Anal..

[12]  Carmen Chicone,et al.  Inertial and slow manifolds for delay equations with small delays , 2003 .

[13]  Kouichi Murakani.  Bifurcated periodic solutions for delayed van der Pol equation , 1999, Neural Parallel Sci. Comput..

[14]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[15]  Mark A. J. Chaplain,et al.  Travelling fronts in a food-limited population model with time delay , 2002, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[16]  Antonio Buonomo.  The Periodic Solution of Van Der Pol's Equation , 1998, SIAM J. Appl. Math..

[17]  Kwok-wai Chung,et al.  Effects of time delayed position feedback on a van der Pol–Duffing oscillator , 2003 .

[18]  Junjie Wei,et al.  Stability and bifurcation analysis in Van der Pol's oscillator with delayed feedback , 2005 .

[19]  Ruihong Li,et al.  Chaos controlling of extended nonlinear Liénard system based on the Melnikov theory , 2006, Appl. Math. Comput..

[20]  J. C. Oliveira,et al.  Oscillations in a van der Pol equation with delayed argument , 2002 .

[21]  S. Ruan.  DELAY DIFFERENTIAL EQUATIONS IN SINGLE SPECIES DYNAMICS , 2006 .

[22]  B. Sandstede,et al.  Fast and Slow Waves in the FitzHugh–Nagumo Equation , 1997 .

[23]  Curtis C. Travis,et al.  On the use of reducible-functional differential equations in biological models , 1982 .