| Project/Area Number |
14340055
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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 |
KOKUBU Hiroshi Kyoto University, Department of Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (50202057)
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| 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)
西田 孝明 京都大学, 大学院・理学研究科, 教授 (70026110)
津田 一郎 北海道大学, 大学院・理学研究科, 教授 (10207384)
岡本 久 京都大学, 数理解析研究所, 教授 (40143359)
深谷 賢治 京都大学, 大学院・理学研究科, 教授 (30165261)
柳田 英二 東北大学, 大学院・理学研究科, 教授 (80174548)
木坂 正史 京都大学, 大学院・人間・環境学研究科, 助教授 (70244671)
石井 豊 京都大学, 大学院・数理学研究院, 助手 (20304727)
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| Project Period (FY) |
2002 – 2004
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| Project Status |
Completed (Fiscal Year 2004)
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| Budget Amount *help |
¥14,100,000 (Direct Cost: ¥14,100,000)
Fiscal Year 2004: ¥3,900,000 (Direct Cost: ¥3,900,000)
Fiscal Year 2003: ¥5,100,000 (Direct Cost: ¥5,100,000)
Fiscal Year 2002: ¥5,100,000 (Direct Cost: ¥5,100,000)
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| Keywords | dynamical system / global structure / bifurcation / chaos / hyperbolicity / ergodic theory / complex dynamics / large degrees of freedom / 特異性 / 不変測度 / 大自由度系 / 特異構造 / 微分方程式 / カオス遍歴 / 摂動 / 複雑系 / 無限次元力学系 / 位相的方法 |
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| 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.
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