Introduction

In the field of engineering, discontinuous dynamical systems are ubiquitous. However, people often tend to use continuous system models to approximate and explore the dynamical characteristics of discontinuous systems for the sake of convenience. Nevertheless, continuous system models often struggle to accurately depict discontinuous systems, potentially leading to problems becoming ambiguous and more complex. This is because discontinuities are omnipresent in discontinuous systems. Therefore, to more precisely describe real-world systems in engineering and perform theoretical analysis and practical predictions, we must employ discontinuous models to characterize these discontinuous systems. Discontinuous dynamical systems can be viewed as composed of numerous continuous subsystems that operate in different domains or at different time intervals. However, the dynamical properties of these subsystems differ significantly from those of interconnected continuous subsystems. Hence, discontinuous dynamical systems can primarily be classified into two categories: the first involves systems with different dynamics in two adjacent time intervals. When one dynamical system reaches a switching point, the system automatically transitions to another dynamical system. The second category pertains to situations where the dynamical systems are distinct in any two neighboring domains. When the flow of subsystems reaches a boundary, it is necessary to provide transmission theorems or transition rules at the boundary. In this book, we will predominantly focus on introducing the fundamental theory of discontinuous dynamical systems. And we will apply this theory to explore the interaction between two dynamical systems. Finally, this chapter will provide an overview of the book’s structure and the content of each chapter.