5. Switching Dynamical Systems

In this chapter, dynamics of switching dynamical systems will be presented. A switching system of multiple subsystems with transport laws at switching points will be discussed. The existence and stability of switching dynamical systems will be discussed through equi-measuring functions. The G-function of the equi-measuring functions will be introduced. The local increasing and decreasing of switching systems to equi-measuring functions will be presented. The global increasing and decreasing of the switching systems to equi-measuring functions will be discussed. Based on the global and local properties of the switching dynamical systems to the equi-measuring function, the stability of switching systems can be discussed. To demonstrate flow regularity and complexity of switching systems, the impulsive system is as a special switching system to present, and the quasi-periodic flows and chaotic diffusion of impulsive systems will be presented. A frame work for periodic flows in switching systems will be presented. The periodic flows and stability for linear switching systems will be discussed. This framework can be applied to nonlinear switching systems. The further results on stability and bifurcation of periodic flows in nonlinear switching systems can be discussed.