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Bifurcations and Chaos in Duffing Equation

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Abstract

The Duffing equation with even-odd asymmetrical nonlinear-restoring force and one external forcing is investigated. The conditions of existence of primary resonance, second-order, third-order subharmonics, morder subharmonics and chaos are given by using the second-averaging method, the Melnikov method and bifurcation theory. Numerical simulations including bifurcation diagram, bifurcation surfaces and phase portraits show the consistence with the theoretical analysis. The numerical results also exhibit new dynamical behaviors including onset of chaos, chaos suddenly disappearing to periodic orbit, cascades of inverse period-doubling bifurcations, period-doubling bifurcation, symmetry period-doubling bifurcations of period-3 orbit, symmetrybreaking of periodic orbits, interleaving occurrence of chaotic behaviors and period-one orbit, a great abundance of periodic windows in transient chaotic regions with interior crises and boundary crisis and varied chaotic attractors. Our results show that many dynamical behaviors are strictly departure from the behaviors of the Duffing equation with odd-nonlinear restoring force.

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Authors and Affiliations

  1. Oriential Scientific and Technologic Institute, Hunan Agriculture University, Changsha, 410081, China

    Meng Zhang

  2. College of Science, China Agricultural University, Beijing, 100083, China

    Jiang-ping Yang

Authors
  1. Meng Zhang
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  2. Jiang-ping Yang
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Correspondence to Meng Zhang.

Additional information

Supported by China Agricultural University (2006062) and by the National Natural Science Foundation of China (No. 10671063).

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Zhang, M., Yang, Jp. Bifurcations and Chaos in Duffing Equation. Acta Mathematicae Applicatae Sinica, English Series 23, 665–684 (2007). https://doi.org/10.1007/s10255-007-0404

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  • DOI: https://doi.org/10.1007/s10255-007-0404

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