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Delay-Induced Double Hopf Bifurcations in a System of Two Delay-Coupled van der Pol–Duffing Oscillators

Heping Jiang

Department of Mathematics, Tongji University, Shanghai 200092, P. R. China

School of Mathematics and Statistics, Huangshan University, Huangshan 245041, P. R. China

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Tonghua Zhang

Department of Mathematics, Swinburne University of Technology, VIC 3122, Australia

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, and

Yongli Song

Department of Mathematics, Tongji University, Shanghai 200092, P. R. China

Author for correspondence.

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https://doi.org/10.1142/S0218127415500583Cited by:19 (Source: Crossref)

Abstract

In this paper, we investigate the codimension-two double Hopf bifurcation in delay-coupled van der Pol–Duffing oscillators. By using normal form theory of delay differential equations, the normal form associated with the codimension-two double Hopf bifurcation is calculated. Choosing appropriate values of the coupling strength and the delay can result in nonresonance and weak resonance double Hopf bifurcations. The dynamical classification near these bifurcation points can be explicitly determined by the corresponding normal form. Periodic, quasi-periodic solutions and torus are found near the bifurcation point. The numerical simulations are employed to support the theoretical results.

Keywords:

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