2000 Fiscal Year Final Research Report Summary
Analysis of dynamical systems and related topics in geometry
| Project/Area Number |
11440054
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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 | Tokyo Institute of Technology |
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Principal Investigator |
ITO Hidekazu Tokyo Institute of Technology, Associate Prof., 大学院・理工学研究科, 助教授 (90159905)
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| Co-Investigator(Kenkyū-buntansha) |
TANAKA Kazunaga Waseda University, Prof., 理工学部, 教授 (20188288)
KOKUBU Hiroshi Kyoto University, Associate Prof., 大学院・理学研究科, 助教授 (50202057)
MORITA Takehiko Tokyo Institute of Technology, Associate Prof., 大学院・理工学研究科, 助教授 (00192782)
NAKAI Isao Ochanomizu University, Prof., 理学部, 教授 (90207704)
ONO Kaoru Hokkaido Univsersity, Prof., 大学院・理学研究科, 教授 (20204232)
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| Project Period (FY) |
1999 – 2000
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| Keywords | Hamiltonian system / integrable system / variational method / zeta function / Conley index / complex dynamical system / symplectic geometry |
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| Research Abstract |
The following is the abstract for the main results obtained under this research project.
1. In the research of Hamiltonian systems, Ito generalized the notion of complete integrability for Hamiltonian systems to that for general vector fields. He proved that the integrability of an analytic vector field is equivalent to the existence of a convergent normalizing transformation near an equlibrium point that are non-resonant and elliptic. It gives an answer to the Poincare center problem.
2. In the research of ergodic theory, Morita studied the zeta function associated with two dimensional scattering billiards problem. He succeeded in extending it meromorphically to a half plane with its real part greater than some negative constant.
3. In the research of bifurcation theory of dynamical systems, Kokubu studied the generalization of Conley index theory to slow-fast systems which are singularly perturbed vector fields. He defined transition matrices when the slow variables are of dimension one, and obtained a general method for proving the existence of periodic or heteroclinic orbits.
4. By using variational method for singular Hamiltonian systems, Tanaka proved the existence of orbits such as (1) scattering type ; (2) periodic orbits under the class of perturbation of type -1/γ^2 ; (3) unbounded and chaotic motions for systems whose potential have two singular points.
5. In the research of symplectic/contact geometry, Ono succeeded in constructing the Floer homology with integer coefficients. Nakai studied 1st order PDE's from the viewpoint of foliation theory and Web geometry. In particular, he defined affine connections for those PDE's with finite type, and used them to study the singularities associated with the foliation defined by their solutions.
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Research Products
(16 results)
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[Publications] Kokubu, H., Mischaikow, K., Oka, H: "Directional transition matrix, in "Conley Index Theory" (Eds.K.Mischaikow, M.Mrozek, P,Zgliczynski)"Banach Center Publication Warsaw, Poland.. 47. 133-144 (1999)
Description
「研究成果報告書概要(欧文)」より
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