2002 Fiscal Year Final Research Report Summary
Global structure of dynamical systems and their bifurcations
| Project/Area Number |
12440048
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Kyoto University |
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Principal Investigator |
NAKAHARA Takako Kyoto Univ., Graduate School of Science, Assistant, 大学院・理学研究科, 助手 (90155797)
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| Co-Investigator(Kenkyū-buntansha) |
MATSUOKA Takashi Naruto Univ. Education College of Education Professor, 学校教育学部, 教授 (50127297)
TSUJII Masato Hokkaido Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (20251598)
SHISHIKURA Mitsuhiro Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70192606)
KOMURO Motomasa Teikyo Univ. of Science Faculty of Science, Associate Professor, メディアサイエンス学科, 助教授 (00186818)
HIRAIDE Koichi Ehime Univ., Faculty of Science, Associate Professor, 理学部, 助教授 (50181136)
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| Project Period (FY) |
2000 – 2002
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| Keywords | dynamical system / giobal structure / bifurcation / complex dynamics / chaos / large degrees of freedom / topological property / invariant measure |
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| Research Abstract |
The results obtained in this research can be classified into four categories: (A) Results on ergodic properties of dynamical systems; (B) Results on topological properties in dynamical systems; (C) Results on large and infinite dimensional dynamical systems; (D) Results on complex dynamical systems. Brief summary of these is given below.
In (A), Tsujii mainly contributed and studied higher dimensional piecewise-expanding maps and partially hyperbolic systems. He obtained some interesting relation and examples on smoothness of piecewise-expanding maps and their invariant measures. He also showed the existence of natural invariant measures for partially hyperbolic systems.
In (B), research was done mainly by Matsuoka and Hiraide, who studied codimention one Anosov diffeomorphisms, and knots, links and braids given by periodic orbits of dynamical systems. Matsuoka studied topological properties of fixed points of homeomorphisms on a compact oriented surface that are isotopic to identity and
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obtained some results on chaos of the maps and stability of fixed points. Shishikura obtained the equivalence between the complete integrability of skew product map based on a irrational rotation of the circle and the existence of a minimal set.
The results in (C) are mainly obtained by Komuro, H. Okamoto (Kyoto Univ.), S. Nii (Kyushu Univ.) and others. Komuro studied a behavior called 'chaotic itenerancy' that is considered typical in systems with large degrees of freedom. He studied this in a globally coupled maps and described its mathematical mechanism in terms of the symmetry of the system and its invariant subspaces.
In (D), Shishikura, M. Kisaka (Kyoto Univ.) and Y. Ishii (Kyushu Univ.) made main contribution. Shishikura studied the Lebesgue measure and the Hausdorff dimension of fractal sets that arise from complex analytic dynamical systems and showed that these quantities change discontinuously through bifurcations of parabolic fixed points. Kisaka obtained several results on structurally finite transcendental entire functions and on wandering domains of such transcendental entire functions. Ishii obtained a condition that determines the hyperbolicity of complex Henon maps, and, on quantum chaos, he made clear the relation between higher quantum tunnel effect and higher dimensional complex dynamics. Less
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