6. Mapping Dynamics and Symmetry

In the previous chapter, dynamics of switching systems were discussed. In this chapter, mapping dynamics and symmetry in discontinuous dynamical systems will be discussed. The G-function of the discontinuous boundary will be presented first. To understand of nonlinear dynamics of a flow from one domain to another domain, mapping dynamics of discontinuous dynamics systems will be presented, which is a generalized symbolic dynamics. Using the mapping dynamics, one can determine periodic and chaotic dynamics of discontinuous dynamical systems, and complex motions can be classified through mapping structure. The nonlinear dynamics of the Chua’s circuit system will be presented as an example. The flow grazing property of a flow will be discussed, which is a source to generate the complex motions in discontinuous dynamical systems. The flow symmetry in discontinuous dynamical systems will be discussed through the mapping dynamics and grazing. The strange attractor fragmentation generated by the grazing of flows to the boundary will be presented.