Periodic Motions in an Autonomous System With a Discontinuous Vector Field

In this paper, parametric characteristics of periodic motions is discussed for a discontinuous dynamical system. Discontinuous dynamical systems extensively exist in engineering, such as switchable circuits, vibro-impact systems, dry-friction systems. A good design of separating boundaries could achieve desired control by letting motions slide along boundaries. Thus the discontinuous dynamical systems were of great interest in past decades. This paper studies a dynamical system consisting of three sub-systems with two circular boundaries. The circular boundaries represent an alike mechanical energy for motions of sub-systems in the two adjacent domains. Owing to the circular boundaries, switching motions of sub-systems in the adjacent domains, sliding motions along boundaries and grazing motions to boundaries can exist in the discontinuous system. Complex periodic motions in such a discontinuous system will be studied. To study complex motions in discontinuous dynamical systems, the sufficient and necessary conditions for motion switchability will be developed through G-functions. The generic mappings in the domains and boundaries are introduced. Based on the conditions of motions switchability at the boundary, mapping structures for specific periodic motions will be developed for periodic motions. Once the periodic motions are determined, the corresponding stability and bifurcations of periodic motions are determined by eigenvalue analysis. Furthermore, the solution of the periodic motions with a grazing point on boundaries is developed by combing the algebraic equations of periodic motions with grazing conditions. The parametric characteristics of periodic motions are stufied through grazing bifurcations and stability switching. A parameter map for excitation frequency versus excitation amplitude is developed. The bifurcation diagrams varying with frequency and amplitude are presented separately to help one better understand the onset and vanishing of periodic motions. Numerical simulations for periodic motions are completed to illustrate the response of discontinuous systems, which are totally different from discontinuous dynamical systems. This study may provide helps in system design and control.

Filippov [1] studied the existence of solutions in discontinuous right-hand side systems. In 1967, Filippov [2] discussed the discontinuous dynamical systems with multi-valued vector fields. Furthermore, Discontinuous dynamical systems were studied systematically in the first book [3]. In 1974, Aizerman and Pyatnisky [4,5] presented a review on the Filippov theory of discontinuous dynamical systems and initially on the sliding mode control. In 1977, Utkin [6] discussed sliding motions along the hyperplane boundary in discontinuous systems. The Filippov theory did not discuss the flow switchability in discontinuous systems, and the grazing and sliding bifurcations were not developed. In 2005, Luo [7] developed a local singularity theory for discontinuous vector fields in dynamical systems. Such a local singularity theory is good for line and plane boundary, and the detailed discussion are presented in Luo [8]. In 2008, Luo [9] developed a theory for flow switchability in discontinuous dynamical systems. The comprehensive discussion on switchability and flow-barrier vector fields in discontinuous dynamical systems were presented in Luo [10]. In 2018, Li and Luo [11] initially discussed the periodic orbit in a discontinuous dynamical system with an elliptic boundary. Herein, such studies will be extended to complex dynamics of a discontinuous dynamical system with two circular boundaries.