| Project/Area Number |
18H01138
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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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| Review Section |
Basic Section 12040:Applied mathematics and statistics-related
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| Research Institution | Tokyo Institute of Technology (2022-2023)
Chubu University (2018-2021) |
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Principal Investigator |
Arai Zin 東京工業大学, 情報理工学院, 教授 (80362432)
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| Co-Investigator(Kenkyū-buntansha) |
石井 豊 九州大学, 数理学研究院, 教授 (20304727)
三波 篤郎 北見工業大学, 工学部, 教授 (30154157)
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2023)
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| Budget Amount *help |
¥17,030,000 (Direct Cost: ¥13,100,000、Indirect Cost: ¥3,930,000)
Fiscal Year 2022: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2021: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2020: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Fiscal Year 2019: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Fiscal Year 2018: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
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| Keywords | 力学系 / カオス / 分岐 / エノン写像 / 計算機援用証明 |
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| Outline of Final Research Achievements |
This study aims to elucidate the mechanisms by which chaotic behavior arises in high-dimensional dynamical systems, thereby contributing to the application of chaos theory. Specifically, we focused on the structure of the parameter space of the Henon map. We found inherently new structures of the set corresponding to connected Julia sets, which can not be achieved by perturbation from one-dimensional theory. This result refines our understanding of the structure of the parameter space of the Henon map, which is a fundamental problem in the study of high-dimensional chaotic dynamical systems. To achieve this goal, we developed several algorithms for computer-assisted proofs, including numerical methods to explore the structure of parameter space efficiently, validated numerical computations of the unstable manifolds of fixed points, and algorithms to evaluate the Green's function and mathematically rigorously prove the existence of its singularities.
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| Academic Significance and Societal Importance of the Research Achievements |
力学系の分岐理論は，カオス理論や数理モデルの解析を通して幅広い分野で応用されいる．しかし、カオスがどのようにして分岐により発生するかに関する数学的な理解は，今だに1次元力学系の解析で得られたものの延長線上にあり，高次元に特有な分岐のストーリーの理解には至っていない．その事が応用でも制約となっている．本研究は，1次元の分岐理論では現れない，高次元カオスの分岐構造を解き明かすための基礎理論やアルゴリズムを整備するものである．現段階ではエノン写像などの基礎的な例の解析にとどまっているが，将来的にこれまでの制約を乗り越えて新しい応用可能性開くための基礎研究であり，社会的意義は高いと思われる．
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