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岔问题，本研究的主要目的是分析最简单的非线性动力系统模型Henon映射的基本性质。当雅可比矩阵等于零时，Henon映射是标准的二次多项式族。因此，分析一维图的性质对于研究Henon图和类Henon图的分岔结构是非常重要的。在直到去年的研究中，我们用拓扑方法分析了一般C^<>单峰映射的1参数族的分岔，并成功地证明了它与标准二次多项式族的分岔是相同的。利用类似的方法，我们可以定义更一般的马蹄形映射的2参数族的周期点分量，并证明了符号序列的一个很自然的充分条件，它表示1-dim - more维部分和双曲部分是如何连通的。R. ghrist证明了R^2上存在一个4次多项式自同构，其悬滞流是泛的，即包含所有连杆类型作为其周期轨道。通过对这个4次多项式自同构的3参数族进行类似的考虑，我们可以给出所有辫的一定的共轭关系。虽然这种关系不是共轭的必要条件，但它的优点是可以很容易地从符号序列的数据中计算出来。这种方法是否能很好地应用于结理论，是一个有待解决的问题。对于保面积Henon映射，期望KAM理论分岔与2符号全移之间存在一定的关系，并且如何将由KAM理论分岔产生的不变圆和周期点嵌入到符号空间中是一个有趣的问题。针对这一问题，我们尝试了几种思路，以便根据Biham-Wenzel方法得到的数值数据得到合适的不变量。然而，我们需要更多的研究来得到一个简洁的数学结果。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