Two Characteristic Behavior Of Piecewise Continuous Systems

In recent years considerable attentions were paid on the discontinuous conservative systems. For example, Bambi Hu, He-Shen Chen and cooperators studied a piecewise continuous map which described the motion of a kicked ion in an infinite potential well. They found some characteristics, such as a stochastic web induced by the discontinuity of the system. Also, Jian Wang and Xiao-Ling Ding found a special property named "quasi-dissipative" in a system concatenated by two area-preserving maps. The system can be viewed as a model of an electronic relaxation oscillator with over- voltage protection. We suggest to study a special kicked rotor moving in a piecewise continuous force field. When adjusting a control parameter the system displays a transition from conservative to quasi-dissipative, and a change of a stochastic web formed by the set of the images of the discontinuous borderlines to a transient stochastic web. This transition and change can be described by the logarithmic dependence of the fractal dimension of the transient stochastic web on the control parameter, or by the variation rule of the averaged lifetime of the iterations in the transient stochastic web when changing the control parameter. These classical characteristics, such as the stochastic web formed by the set of the images of the discontinuous borderlines and the quasi-dissipative property, are interesting. An advantage of the current system is the possibility for searching their quantum correspondences. This thesis reports our first step investigation on it.Another system investigated is an electronic relaxation oscillator. A new kind of crisis, which is marked by a sudden change of a strange repeller, is observed in it. In order to analytically deduce the characteristic scaling law, we have constructed a simplified piecewise linear model that describes the characteristic phenomenon so that we can quantitatively and analytically deduce the sudden change of the rules of the fractal dimension of the strange repeller and the averaged lifetime in the region occupied by the original attractor at a critical parameter value when the repeller disappears. Our numerical investigation in the original electronic relaxation oscillatorshows very good agreement with the analytical conclusions.