In this paper, a local theory of non-smooth dynamical systems on connectable and accessible sub-domains is developed. The properties for separation boundaries based on the characteristics of flows are determined, and the sliding dynamics on a specified separation boundary is introduced. The local singularity and transversality of a flow on the separation boundary from a domain into its adjacent domains are investigated, and the bouncing and tangency of the flows to the separation boundary for non-smooth dynamical systems are discussed as well. The sufficient and necessary conditions for the local singularity, transversality and bouncing of the flows are developed. These conditions are applicable for determining complicated dynamical behaviors of non-smooth dynamical systems. Ó 2004 Elsevier B.V. All rights reserved. PACS: 05.45.)a; 45.20.)d

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The dynamical behavior of a bouncing ball with a sinusoidally vibrating table is revisited in this paper. Based on the equation of motion of the ball, the mapping for period-1 motion is constructed and thereby allowing the stability and bifurcation conditions to be determined. Comparison with Holmes's solution [1] shows that our range of stable motion is wider, and through numerical simulations, our stability result is observed to be more accurate. The Poincar6 mapping sections of the unstable period-1 motion indicate the existence of identical Smale horseshoe structures and fractals. For a better understanding of the stable and chaotic motions, plots of the physical motion of the bouncing ball superimposed on the vibration of the table are presented.

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