The structure and the bifurcation of low dimensional non-linear dynamical systems.

点击翻译按钮获取中文摘要

In order to investigate the structure and the bifurcation of discrete non-linear dynamical systems, the main purpose we have in this research is to analyze the basic properties of the Henon map which is the simplest model of non-linear dynamical systems.When the Jacobian is equal to zero, the Henon map is the standard family of quadratic polynomials. Therefore, the analysis of the properties of 1-dimensional maps is very important for the study of the bifurcation structure of the Henon map and Henon like maps. In the research until the last year, we analysed the bifurcaion of 1-parameter families of general C^<> unimodal maps by a topological approach, and succeeded to prove that it was the same as that of the standard family of quadratic polynomials. By applying the similar method, we can define periodic point components for 2-parameter families of more general horseshoe like maps, and can prove a quite natural sufficient condition for symbolic sequences which represents how the 1-dim … More ensional parts and the hyperbolic parts are connected.R.Ghrist have proved that there existed a polynomial automorphism of degree 4 on R^2 whose suspension flow was universal, namely, it contains all link types as its periodic orbits. By making the similar consideration on the 3-parameter family of this polynomial automorphism of degree 4, we can give a certain conjugacy relation for all braids. Although this relation is not a necessary condition for the conjugacy, it has an advantage that it can be calculated easily from the data of symbolic sequences. It is a future question whether this method has a good application to the knot theory.For area preserving Henon map, a certain relation between KAM theoretic bifurcation and 2-symbol full shift is expected, and it is an interesting question how invariant circles and periodic points arising from KAM theoritic bifurcaion are embeded in the symbol space. On this problem, we tried several ideas in order to get an appropriate invariant based on numerical data obtained by the Biham-Wenzel method. However, we need more investigation to get a neat mathematical result. Less