On Flow Barriers and Switchability in Discontinuous Dynamical Systems

In this paper, the theory of flow barriers in discontinuous dynamical systems is systematically presented as a new theory for the first time, which helps one rethink the existing theories of stability and control in dynamical systems. The concept of flow barriers in discontinuous dynamical systems is introduced, and the passability of a flow to the separation boundary with flow barriers is presented. Because the flow barriers exist on the separation boundary, the switchability of a flow to such a separation boundary is changed accordingly. The coming and leaving flow barriers in passable flows are discussed first, and the necessary and sufficient conditions for a flow to pass through the boundary with flow barrier are developed. Flow barriers for sink and source flows are also discussed. Once the sink flow is formed, the boundary flow will exist. When the boundary flow disappears from the boundary, the boundary flow barrier on the boundary may exist, which is independent of vector fields in the corresponding domains. Thus, the necessary and sufficient conditions for formations and vanishing of the boundary flow are developed. A periodically forced friction model is presented as an example for a better understanding of flow barrier existence in physical problems. The flow barrier theory presented in this paper may provide a theoretic base to further develop control theory and stability.