2021 Fiscal Year Final Research Report
Reserarch on integrable structure of dynamical systems by geometric methods
| Project/Area Number |
17K05271
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Tokyo University of Marine Science and Technology |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2022-03-31
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| Keywords | パンルヴェ方程式 / 力学系 / 初期値空間 / 対称性 |
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| Outline of Final Research Achievements |
The purpose of this study is to clarify various properties of discrete dynamical systems through actions on the geometric structure of phase spaces. As a concrete result, we found a general formula for constructing bi-rational maps from abstract actions on Picard groups, and established a method of deautonomization that preserves integrable structures. We also established a method for constructing a phase space (initial value space) in which the action is appropriately represented for higher dimensional dynamical systems. For higher dimensional dynamical systems that are not integrable, we found a way to construct algebraically stable manifolds. Furthermore, we proposed a natural solution construction method for the Quispel-Roberts-Thompson map, which is a self-isomorphic map of elliptic surfaces.
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| Free Research Field |
力学系
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| Academic Significance and Societal Importance of the Research Achievements |
力学系には時間変数に関して連続的なものと離散的なものがあるが，連続のものは離散的なものの極限として得られることから，離散的なものの方がより一般的である．離散力学系の性質を調べる有力手段の一つにそれが自然に作用する初期値空間と呼ばれる多様体を構成し，その多様体の性質を調べるという方法がある．さらに多様体については代数幾何等の道具を使って性質を調べることができる．本研究はこのような方向で進めたものであり，特に高次元の場合や，可積分なときについて新たな手法を提案した．
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