2004 Fiscal Year Final Research Report Summary
Bifurcation theoretical approach to chaotic dynamics and to systems with large degrees of freedom
| Project/Area Number |
14340055
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Global analysis
|
|---|
| Research Institution | KYOTO UNIVERSITY |
|---|
Principal Investigator |
KOKUBU Hiroshi Kyoto University, Department of Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (50202057)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
SHISHIKURA Mitsuhiro Kyoto University, Department of Mathematics, Professor, 大学院・理学研究科, 教授 (70192606)
ASAOKA Masayuki Kyoto University, Department of Mathematics, Lecturer, 大学院・理学研究科, 講師 (10314832)
ARAI Zin Kyoto University, Department of Mathematics, Assistant Professor, 大学院・理学研究科, 助手 (80362432)
NISHIURA Yasumasa Hokkaido University, Research Institute of Electronic Science, Professor, 電子科学研究所, 教授 (00131277)
TSUJII Masato Hokkaido University, Department of Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (20251598)
|
|---|
| Project Period (FY) |
2002 – 2004
|
| Keywords | dynamical system / global structure / bifurcation / chaos / hyperbolicity / ergodic theory / complex dynamics / large degrees of freedom |
|---|
| Research Abstract |
Global structures and bifurcations of dynamical systems, with special emphasis on chaos-complicated and unpredictable behavior in dynamics-and systems of large degrees of freedom such as PDEs and coupled systems, are studied from various different points of view and many interesting results are obtained. As some of main results in this project, Kokubu (1)showed the existence of a singular invariant set called "singularly degenerate heteroclinic cycle" in the Lorenz system and its alike, from which a chaotic attractor of geometric Lorenz type is proven to bifurcate, (2)developed a theory describing the structure of singularly perturbed vector fields with using a topological invariant called Conley index, obtained a method to show the existence of periodic and chaotic solutions in such systems under suitable setting, and applied it to several concrete problems. Shishikura studied complex analytic dynamical systems, and in particular developed a renormalization theory for parabolic fixed points, which will be a new and very powerful tool for studying the structure and bifurcation of such systems. Asaoka studied dynamical systems with a sort of hyperbolicity called projectively Anosov structure and completed a classification in the case of 3-dimensional flows. Combining rigorous computation with topological methods such as the Conley index theory, Arai obtained several interesting results on hyperbolicity and global bifurcations in the Henon maps. Tsujii studied dynamical systems from ergodic theory viewpoint and obtained a general result on the existence of good invariant measures in 2-dimensional partially hyperbolic systems. Nishiura studied complicated interesting transient behavior observed in some kinds of PDEs called self-replicating and self-destruction patterns and clarified its mechanism by using dynamical system theory. Other results on systems with large degrees of freedom include Komuro's detailed analysis on chaotic itenerancy in globally coupled maps.
|
