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Homoclinic bifurcation in a quasiperiodically excited impact inverted pendulum

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Abstract

Homoclinic bifurcation for a nonlinear inverted pendulum impacting between two rigid walls under external quasiperiodic excitation is analyzed. The results for the homoclinic bifurcation of quasiperiodically excited smooth systems obtained by Ide and Wiggins are extended to the non-smooth ones. We present a method of Melnikov type to derive sufficient conditions under which the perturbed stable and unstable manifolds intersect transversally. Such a transversal Intersection implies the appearance of Smale horseshoe-type chaotic dynamics that is similar to that in the periodically forced smooth systems. As an application, by using a combination of analytical and numerical methods, a quasiperiodically excited impact oscillator of Duffing type with two frequencies is studied in detail.

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Acknowledgments

We are very grateful to the editor and the referees for their careful checking and helpful comments that have notably improved the paper. This work is supported by National Natural Science Foundation of China under Grant No. 11371264.

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  1. Department of Mathematics, Sichuan University, Chengdu, 610064, Sichuan, People’s Republic of China

    Junming Gao & Zhengdong Du

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  1. Junming Gao
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  2. Zhengdong Du
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Correspondence to Zhengdong Du.

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Gao, J., Du, Z. Homoclinic bifurcation in a quasiperiodically excited impact inverted pendulum. Nonlinear Dyn 79, 1061–1074 (2015). https://doi.org/10.1007/s11071-014-1723-4

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