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Analytical period-3 motions to chaos in a hardening Duffing oscillator

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Abstract

In this paper, bifurcation trees of period-3 motions to chaos in the periodically forced, hardening Duffing oscillator are investigated analytically. Analytical solutions for period-3 and period-6 motions are used for the bifurcation trees of period-3 motions to chaos. Such bifurcation trees are based on the Hopf bifurcations of asymmetric period-3 motions. In addition, an independent symmetric period-3 motion without imbedding in chaos is discovered, and such a symmetric period-3 motion possesses saddle-node bifurcations only. The switching of symmetric to asymmetric period-3 motions is completed through saddle-node bifurcations, and the onset of asymmetric period-6 motions occurs at the Hopf bifurcations of asymmetric period-3 motions. Continuously, the onset of period-12 motions is at the Hopf bifurcation of asymmetric period-6 motions. With such bifurcation trees, the chaotic motions relative to asymmetric period-3 motions can be determined analytically. This investigation provides a systematic way to study analytical dynamics of chaos relative to period-m motions in nonlinear dynamical systems.

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Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL, 62026-1805, USA

    Albert C. J. Luo & Jianzhe Huang

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  1. Albert C. J. Luo
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  2. Jianzhe Huang
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Correspondence to Albert C. J. Luo.

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Luo, A.C.J., Huang, J. Analytical period-3 motions to chaos in a hardening Duffing oscillator. Nonlinear Dyn 73, 1905–1932 (2013). https://doi.org/10.1007/s11071-013-0913-9

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