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

为了研究离散非线性动力系统的结构和分叉，本研究的主要目的是分析作为最简单的非线性动力系统模型的Henon映射的基本性质，当雅可比等于零时，Henon映射是标准的二次多项式族。因此，分析一维映射的性质对于研究Henon映射和Henon-like映射的分支结构是非常重要的。在过去一年的研究中，我们用拓扑方法分析了一般单峰映射族的单参数分支，并成功地证明了它与标准二次多项式族的分支是相同的。应用类似的方法，我们可以定义更一般的马蹄形映射族的周期点分量，并且可以证明符号序列的一个相当自然的充分条件，它表示了1维…R.Ghrist证明了R^2上存在一个4次多项式自同构，它的悬浮流是普适的，即它包含所有的链环类型作为其周期轨道.通过对这种四次多项式自同构的三参数族进行类似的考虑，我们可以给出所有辫子的某种共轭关系。虽然这种关系不是共轭的必要条件，但它的一个优点是它可以很容易地从符号序列的数据中计算出来。对于保面积Henon映射，KAM理论分支与2-符号全移位之间存在一定的联系，而由KAM理论分支产生的不变圆和周期点如何嵌入符号空间是一个有趣的问题。在这个问题上，我们尝试了几种想法，以获得一个基于Biham-Wenzel方法得到的数值数据的合适的不变量。然而，我们还需要更多的调查才能得到一个简洁的数学结果。较少

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