Dissipation, Semi-dissipative And The Upheaval. Class

The aim of this thesis is to introduce some characteristic phenomena in discontinuous and non-invertible two-dimensional maps, which may be a concatenation of two conservative or dissipative maps. The first chapter introduces some basic conceptions. The second chapter introduces the studies on some systems already reported by our group: an electronic circuit with over-voltage protection investigated by Wang J et al.; a kicked rotor in piecewise continuous force field investigated by Jiang Y M et al., and another circuit discussed by Chao X G et al. The third chapter discusses a kicked billiard model, with the controlling parameter changing, the system transits from conservation to quasi-dissipation, and from continuous to discontinuous. It makes a fat-fractal conservative stochastic web change to a transient chaotic stochastic web, and iterations on the transient web escape to some elliptic islands through leaking holes, which is the intersection set of the elliptic islands with the set having two back images. The sudden change of stochastic web can be considered as a crisis. We calculate the lifetime scaling exponent numerically and also present a qualitative discussion analytically. We also discuss the fat-fractal forbidden web caused by discontinuity and non-invertibility and the scaling rule describing its evolution. The forth chapter reports another electronic circuit. When the controlling parameter changes, the system transfers from a semi-dissipative system to a discontinuous dissipative one. In the semi-dissipative system, there are chaotic attractors and elliptic islands on the phase space. With the parameter changing, chaotic attractor and the elliptic islands disappear, and two point periodical attractors appear. This again can be considered as a kind of crisis. We discuss the following characters in the system: lifetime scaling exponent, forbidden web and the basins of attraction. The last chapter concludes the work about the discontinuous and non-invertible two-dimensional maps, and showed that the quasi-dissipation is a real phenomenon no matter how the Poincare section is chosen. In this chapter, we also discussed some important common characters in the system: the fat fractal forbidden web, and the riddle-like basin caused by it if there are different attractors coexist. When the strong dissipation works, the basin of the attraction in strong dissipative area, with regular or fractal border, often becomes the leaking hole in a crisis. At last, we give a example which explains the fake phenomena caused by changing the Poincare sections periodically when time divelopes. To some extent, this thesis can be seen as the summarization of our group's work in recent years.