2012 Fiscal Year Final Research Report
Study of Global Structures and Bifurcations of Dynamical Systems including Systems with Large Degrees of Freedom
| Project/Area Number |
21340035
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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 京都大学, 大学院・理学研究科, 教授 (50202057)
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| Co-Investigator(Renkei-kenkyūsha) |
ARAI Zin 北海道大学, 大学院・理学研究院, 准教授 (80362432)
OKA Hiroe 龍谷大学, 理工学部, 教授 (20215221)
OBAYASHI Ippei 京都大学, 大学院・理学研究科, 特定研究員 (30583455)
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| Project Period (FY) |
2009 – 2012
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| Keywords | 力学系 / 分岐 / 大域的構造 / 不変集合 / 計算機援用証明 |
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| Research Abstract |
A great improvement of the Conley-Morse graph method, a computer-assisted method that analyzes global dynamics and bifurcations by combining a topological method and validated numerical computation has been done by developing a computer-algorithm for generating so-called clutching graphs which describe relation between Conley-Morse graphs on adjacent parameter domains. Another improvement is a method of non-uniform grid decomposition of the phase space in order to substantially decrease computational cost of the Conley-Morse graph method.A new approach to the bifurcation theory of dynamical systems from topological-computation theory viewpoint has been proposed which captures bifurcation of dynamics from changes of the corresponding Conley-Morse graphs. From this viewpoint, we have obtained a new mathematical framework for the crisis bifurcation.Moreover, we have studied the global dynamics of a Coupled Map Lattice system and a Coupled Oscillator system by using the Conley-Morse graph method, and have obtained the creation and bifurcations of some global dynamics such as unstable invariant tori and their connecting orbits.
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