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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

A Parameter Study on Periodic Motions in a Discontinuous Dynamical System with Two Circular Boundaries

Discontinuity, Nonlinearity, and Complexity 10(2) (2021) 289--309 | DOI:10.5890/DNC.2021.06.009

Siyu Guo, Albert C. J. Luo

Department of Mechanical and Mechatronics Engineering, Southern Illinois University Edwardsville, Edwardsville, IL62026-1805, USA

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Abstract

In this paper, periodic motions in a discontinuous dynamical system with two circular boundaries are studied analytically by generic mappings. A bifurcation tree of stable and unstable periodic motions varying with excitation frequency is predicted analytically. On the bifurcation tree, there are three main bifurcations: the grazing bifurcation for the motions switching, the period-doubling bifurcations for period-doubled periodic motion, and saddle-node bifurcations for onset and vanishing of periodic motions. Periodic motions are numerically illustrated, and the $G$-functions are presented for illustrations of the analytical conditions of motions switchability, such as, the passable motion and grazing motion at the boundaries, and the formation and vanishing of sliding motions on the discontinuous boundaries. In this study discussed are the parameter effects on periodic motions in discontinuous dynamical systems. Such discussion is very helpful for one to design a discontinuous system for specific motions under specific system parameters.

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