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Discontinuity, Nonlinearity, and Complexity 15(1) (2026) 121--144 | DOI:10.5890/DNC.2026.03.009

Albert C J Luo, Tianji Ma

Department of Mechanical and Mechatronics Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA

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Abstract

In this paper, impulsive periodic motions in a pendulum under sinusoidal impulses are studied through the implicit mapping method. From the specific mapping structures, the periodic motions in the sinusoidally impulsive pendulum are obtained, and the stability and bifurcation of the corresponding impulsive periodic motions are analyzed. The bifurcation trees of the impulsive periodic motions to chaos are presented, and the impulsive homoclinic orbits are also obtained. The impulsive homoclinic orbits are for the homoclinic bifurcations of the impulsive periodic motions. The impulsive periodic motions and impulsive homoclinic orbits of the pendulum under sinusoidal impulses are illustrated. Such a method can be extended to the switching system with transport laws.

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