Introduction

Discontinuous dynamical systems exist everywhere in the real world. One used to adopt continuous models for approximate descriptions of discontinuous dynamical systems. However, such continuous modeling cannot provide adequate predictions of discontinuous dynamical systems, and also makes the problems solving be more complicated and inaccurate. In the real world, the discontinuous modeling of dynamical systems is absolute, but the continuous modeling is relative. In other words, the continuous description of dynamical systems is an approximation of the discontinuous problems. To better describe the real world, one should realize that discontinuous models can provide adequate and real predications of engineering systems. For any discontinuous dynamical system, there are many continuous subsystems in different domains or different time intervals, and the dynamical properties of any continuous subsystems are different from that of the adjacent continuous subsystems. Thus, we have two types of discontinuous dynamical systems. (i) In two different adjacent time intervals, dynamical systems are different. When a dynamical system reaches the switching time, this system will be switched to another different dynamical system. With such switching, the discontinuous dynamical system is called the switching system. (ii) In phase space, there are many different domains. On any two adjacent domains, distinct dynamical systems are defined. Thus, once a flow of a sub-system arrives to its boundary, the switchability and/or transport laws on the boundary should be addressed because of such a difference between two adjacent subsystems.