Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed Delays | Zendy

Xiaobing Zhou | Zendy; Murong Jiang | Zendy; Xiaomei Cai | Zendy

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Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed Delays

Author(s) -

Xiaobing Zhou,

Murong Jiang,

Xiaomei Cai

Publication year - 2011

Publication title -

discrete dynamics in nature and society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.264

H-Index - 39

eISSN - 1607-887X

pISSN - 1026-0226

DOI - 10.1155/2011/569141

Subject(s) - mathematics , transcendental equation , center manifold , characteristic equation , hopf bifurcation , van der pol oscillator , mathematical analysis , bifurcation , stability (learning theory) , numerical analysis , differential equation , physics , nonlinear system , computer science , quantum mechanics , machine learning

We consider the van der Pol equation with discrete and distributed delays. Linear stability of this equation is investigated by analyzing the transcendental characteristic equation of its linearized equation. It is found that this equation undergoes a sequence of Hopf bifurcations by choosing the discrete time delay as a bifurcation parameter. In addition, the properties of Hopf bifurcation were analyzed in detail by applying the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to illustrate and verify the theoretical analysis. Copyright © 2011 Xiaobing Zhou et al.

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