The structure and the bifurcation of low dimensional nonlinear dynamical systems.
Project/Area Number  09640083 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Geometry

Research Institution  Kitami Institute of Technology 
Principal Investigator 

CoInvestigator(Kenkyūbuntansha) 
辻井 正人 北海道大学, 理学部, 助教授 (20251598)
河野 正晴 北見工業大学, 工学部, 教授 (40170203)
KOUNO Masaharu
TSUJII Masato

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 1998 : ¥1,200,000 (Direct Cost : ¥1,200,000)
Fiscal Year 1997 : ¥1,800,000 (Direct Cost : ¥1,800,000)

Keywords  nonlinear / dynamical Systems / chaos / bifurcation / Henon map / unimodal map / braid / periodic point / 非線形 / 力学系 / カオス / 分岐 / broid / 周期点 / dynamical system / symbolic dynamics 
Research Abstract 
In order to investigate the structure and the bifurcation of discrete nonlinear 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 nonlinear 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 1dimensional 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 1parameter 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 2parameter families of more general horseshoe like maps, and can prove a quite natural sufficient condition for symbolic sequences which represents how the 1dim
… 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 3parameter 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 2symbol 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 BihamWenzel method. However, we need more investigation to get a neat mathematical result. 面積を保存するHenon familyにおいては,2symbolのfull shiftと,KAM理論的分岐との開連が予想され,これによって発生する周期点やinvariant circleなどが,どのようにsymbolic dynamicsの中に埋めこまれているのか?という興味深い問題がある。これについては,前年度にひき続き,BihamWenzelの方法による数値実験で得られたデータをもとに,適切な不変量を定義するための試みをいくつか行なったが,まとまった理論的結果とするためには,さらなる研究が必要である。 Less

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Research Output
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