Switchability and Attractivity of Domain Flows

Abstract

In this chapter, the switchability and attractivity of domain flows to edges of domains will be discussed. The classification and definition of edges will be presented, and the corresponding dynamical systems on domains, boundaries, edges, and vertexes will be defined. The coming, leaving and tangency of a domain flow to a specific edge will be discussed through the corresponding boundaries. The switchability and passability of a flow from an accessible domain to another accessible domain will be presented with a switching rule. The convex and concave edges are introduced for discontinuous dynamical systems. In addition, the mirror domains will also be introduced through the extension of boundaries at the convex edge. The transversally grazing passability of a flow to the concave edges will be presented. The equi-measuring surface will be introduced, and the attractivity of a domain flow to the boundary will be discussed. Further, an equi-measuring edge in domain will be presented, and the attractivity of a domain flow to a specific edge will be discussed. The bouncing domain flows to a specific edge will be discussed for multi-valued vector fields.