1999 Fiscal Year Final Research Report Summary
Topological indices of dynamical systems
| Project/Area Number |
09440043
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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 |
Geometry
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| Research Institution | Ryukoku University |
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Principal Investigator |
YAMAGISHI Yoshikazu Ryukoku University, Department of Applied Mathematics and Informatics, Instructor, 理工学部, 助手 (40247820)
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| Co-Investigator(Kenkyū-buntansha) |
USHIKI Shigehire Kyoto University, Graduate School of Human and Environmental Studies, Professor, 大学院・人間環境学研究科, 教授 (10093197)
KOKUBU Hiroe (OKA) Ryukoku University, Department of Applied Mathematics and Informatics, Professor, 理工学部, 教授 (20215221)
ITO Toshikazu Ryukoku University, Department of Applied Mathematics and Informatics, Professor, 経済学部, 教授 (60110178)
NII Shunsaku Saitama University, Department of Mathematics, Research Associate, 理学部, 助手 (50282421)
MATSUOKA Takashi Naruto University of Education, College of Education, Associate Professor, 学校教育学部, 助教授 (50127297)
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| Project Period (FY) |
1997 – 1999
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| Keywords | topological index / periodic indeterminate point / Ruelle's transfer operator / homoclinic orbit / fixed point index / bifurcation structure of traveling waves / surface homeomorphism / stability |
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| Research Abstract |
In this project various kinds of problems which arise in dynamical systems (defied as iteration of maps or vector fields on a manifold) are studied using several types of topological methods, which resulted in the following :
1. In complex dynamical systems arising from the Newton's method for polynomial equations with 2 variables, the geometric structure around a periodic indeterminate point is investigated by constructing a Cantor set family ;of invariant manifolds.
2. Using a topological invariant of dynamical systems called Conley index, an axiomatic definition of the transition matrix and its generalization to higher parameter dimensions is given, which describes changes of the global structure of dynamical systems caused by global orbits such as homoclinic as well as heteroclinic orbits.
3. Topological structure of periodic point sets of planar maps is studied using the braid invariant and the fixed point index, and an estimate of the number of periodic points and their stability are described using these invariants.
4. Bifurcation structure of traveling waves are described in terms of a topological invariant called the stability index, and a method for determining the stability of traveling waves from the bifurcation information is obtained.
5. An improved deformation algorithm using graphs is obtained for constructing canonical forms of surface homoeomorphism. Topology of branched surfaces that admit expanding immersions is also studied.
6. A new approach for studying complex dynamical systems using Ruelle's transfer operators is developed.
7. Several applications of Poincare-Bendixson type theorems for holomorphic vector fields are obtained.
These results clearly show the effectiveness of the topological methods in the study of dynamical systems, and such an approach will become even more important in the future research.
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