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Bifurcation trees of period-1 motions in a periodically excited, softening Duffing oscillator with time-delay

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A Correction to this article was published on 07 June 2019

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Abstract

In this paper, bifurcation trees of period-1 motions to chaos in a periodically excited, time-delay, softening Duffing oscillator are analytically predicted through an implicit mapping method. Discretization of the time-delay oscillator gives an implicit mapping. Stable and unstable periodic motions in such a time-delay, softening Duffing oscillator are achieved through the corresponding mapping structures. From the finite discrete Fourier series, harmonic frequency–amplitude characteristics for stable and unstable solutions of period-1 to period-4 motions are developed, and the singularity, catastrophes and quantity levels of harmonic amplitudes are presented. A symmetric period-1 motion with symmetric break generates three branches of period-1 motions to chaos. From the analytical prediction, periodic motions in the time-delay softening Duffing oscillator are simulated numerically. The bifurcation trees of period-1 motions to chaos in the time-delay softening Duffing oscillator are difficult to be obtained from the traditional analytical methods.

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Change history

  • 07 June 2019

    In the original publication, Fig. 1a, b was published incorrectly. The corrected figure is given below.

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Authors and Affiliations

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

    Siyuan Xing & Albert C. J. Luo

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  1. Siyuan Xing
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  2. Albert C. J. Luo
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Correspondence to Albert C. J. Luo.

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Xing, S., Luo, A.C.J. Bifurcation trees of period-1 motions in a periodically excited, softening Duffing oscillator with time-delay. Int. J. Dynam. Control 7, 842–855 (2019). https://doi.org/10.1007/s40435-019-00520-1

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  • DOI: https://doi.org/10.1007/s40435-019-00520-1

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