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

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  Kitami Institute of Technology 
Principal Investigator 

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

Project Period (FY) 
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 / braid / dynamical system / bifurcation / chaos / periodic point / 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. Less

Report
(3results)
Research Output
(4results)