Neimark bifurcations of a generalized duffing-van der pol oscillator with nonlinear fractional order damping

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  • A. Y T Leung
  • H. X. Yang
  • P. Zhu


Original languageEnglish
Article number1350177
Journal / PublicationInternational Journal of Bifurcation and Chaos
Issue number11
Publication statusPublished - Nov 2013


A generalized Duffing-van der Pol oscillator with nonlinear fractional order damping is introduced and investigated by the residue harmonic homotopy. The cubic displacement involved in fractional operator is used to describe the higher-order viscoelastic behavior of materials and of aerodynamic damping. The residue harmonic balance method is employed to analytically generate higher-order approximations for the steady state responses of an autonomous system. Nonlinear dynamic behaviors of the harmonically forced oscillator are further explored by the harmonic balance method along with the polynomial homotopy continuation technique. A parametric investigation is carried out to analyze the effects of fractional order of damping and the effect of the magnitude of imposed excitation on the system using amplitude-frequency curves. Jump avoidance conditions are addressed. Neimark bifurcations are captured to delineate regions of instability. The existence of even harmonics in the Fourier expansions implies symmetry-breaking bifurcation in certain combinations of system parameters. Numerical simulations are given by comparing with analytical solutions for validation purpose. We find that all Neimark bifurcation points in the response diagram always exist along a straight line. © 2013 World Scientific Publishing Company.

Research Area(s)

  • Generalized fractional Duffing-van der Pol oscillator, Neimark bifurcations, Nonlinear dynamic behavior, Polynomial homotopy continuation, Residue harmonic balance method