Periodic motions and homoclinic orbits in a discontinuous dynamical system on a single domain with multiple vector fields

Abstract

In this paper, periodic motions and homoclinic orbits in a discontinuous dynamical system on a single domain with two vector fields are discussed. Constructing periodic motions and homoclinic orbits in discontinuous dynamical systems is very significant in mathematics and engineering applications, and how to construct periodic motions and homoclinic orbits is a central issue in discontinuous dynamical systems. Herein, how to construct periodic motions and homoclinic orbits is presented through studying a simple discontinuous dynamical system on a domain confined by two prescribed energies. The simple discontinuous dynamical system has energy-increasing and energy-decreasing vector fields. Based on the two vector fields and the corresponding switching rules, periodic motions and homoclinic orbits in such a simple discontinuous dynamical system are studied. The analytical conditions of bouncing, grazing, and sliding motions at the two energy boundaries are presented first. Periodic motions and homoclinic orbits in such a discontinuous dynamical system are determined through the specific mapping structures, and the corresponding stability is also presented. Numerical illustrations of periodic motions and homoclinic orbits are given for constructed complex motions. Through this study, using discontinuous dynamical systems, one can construct specific complex motions for engineering applications, and the corresponding mathematical methods and computational strategies can be developed.