A theory for non-smooth dynamic systems on the connectable domains

In this paper, a local theory of non-smooth dynamical systems on connectable and accessible sub-domains is developed. The properties for separation boundaries based on the characteristics of flows are determined, and the sliding dynamics on a specified separation boundary is introduced. The local singularity and transversality of a flow on the separation boundary from a domain into its adjacent domains are investigated, and the bouncing and tangency of the flows to the separation boundary for non-smooth dynamical systems are discussed as well. The sufficient and necessary conditions for the local singularity, transversality and bouncing of the flows are developed. These conditions are applicable for determining complicated dynamical behaviors of non-smooth dynamical systems.