This presentation will talk about the recent development of the mapping structure of regular and chaotic motions in non-smooth dynamic systems. The boundary sets and singular sets for such non-smooth systems are introduced, and transport laws on the discontinuous boundary sets will be discussed. Further, switching planes and basic mappings will be developed. Based on those mappings, the mapping structure of regular and chaotic motions will be constructed, and the corresponding stable and unstable solutions of regular motions can be obtained. The transition mechanism of a periodic motion to its adjacent periodic motion with the same mapping complexity is discussed through bifurcation, chaos, and catastrophe and grazing. Consider a forced, piecewise linear system as an example to demonstrate such a theory. Analytical solutions for possible periodic motions with low combination of mappings are presented for illustration. The strange attractors with certain mapping structures are illustrated. This theory is applicable to linear and nonlinear discontinuous systems for periodic solution, stability, bifurcation and chaos.
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