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Suppression of chaos in a generalized Duffing oscillator with fractional-order deflection

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Abstract

A generalized Duffing oscillator with fractional-order deflection can be used to model the oscillatory motion of a buckled beam with simply supported or hinged ends. In this work, the problem of suppression of chaos in such a Duffing oscillator is considered. We show the appropriate range of parameters for the control of horseshoe chaos by introducing external periodic resonant excitation and parametric excitation as chaos-suppressing perturbation. Through the Melnikov technique, we obtain that in addition to the frequency, the phase difference between the chaos-inducing excitation and the chaos-suppressing excitation of systems plays a key role in chaos suppression. Given the optimum phase that satisfies the inhibition theorems, we compare the chaos-suppressing efficiency of external and parametric periodic perturbations for the principal resonance case. Compared with parametric (external) excitation, external (parametric) excitation with a frequency above (below) a critical value is more effective in suppressing homoclinic chaos because it provides a wider amplitude range. The results hold for an arbitrary deflection order as either an integer or a fraction, which depends on the material and bending properties of the beam, as long as its value is larger than 1. Moreover, the critical value of the frequency will shift to a larger value as the deflection order increases.

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Acknowledgements

The work was supported by the National Natural Science Foundation of China (Nos. 11672233, 11672231 and 11672230), the NSF of Shaanxi Province (No. 2016JM1010) and the NPU Foundation for Fundamental Research (No. 3102017AX008). We appreciate constructive comments and suggestions of the reviewers for great help on this work.

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  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, China

    Lin Du, Yunping Zhao, Youming Lei, Jian Hu & Xiaole Yue

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  1. Lin Du
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  2. Yunping Zhao
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  4. Jian Hu
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  5. Xiaole Yue
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Correspondence to Lin Du.

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Du, L., Zhao, Y., Lei, Y. et al. Suppression of chaos in a generalized Duffing oscillator with fractional-order deflection. Nonlinear Dyn 92, 1921–1933 (2018). https://doi.org/10.1007/s11071-018-4171-8

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