Imaginary, sink and source flows in the vicinity of the separatrix of non-smooth dynamic systems

In this letter, the real and imaginary flows for non-smooth dynamical systems are described, and the δ-layer of the separation boundary is introduced as well. The onset, existence and vanishing of the sink and source flows in the δ-layer are presented. The switching between the two semi-passable flows and the switching between the sink and source flows are investigated through the singular gluing points. Finally, the necessary and sufficient conditions for the onset, vanishing and switching are presented. These conditions provide the criteria to determine the sliding motion on the separatrix, which can be very easily applied to non-smooth dynamical systems.