Development of rigorous computation methods for singular trajectories in dynamical systems
| Project/Area Number |
17K14235
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Foundations of mathematics/Applied mathematics
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| Research Institution | Kyushu University |
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Principal Investigator |
Matsue Kaname 九州大学, マス・フォア・インダストリ研究所, 助教 (70610046)
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| Project Period (FY) |
2017-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
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| Keywords | 特異構造の一様評価 / コンパクト化 / 時間スケール特異性解消 / 爆発レート / 精度保証付き数値計算 / 力学系 / fast-slow系 / 爆発解 / 燃焼 / 絶滅進行波 / コンパクトン進行波 / 漸近挙動 / 応用数学 / 数値解析 |
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| Outline of Final Research Achievements |
I have developed methodologies to describe solution structures of differential equations possessing singular perturbation nature, including "multi-scale structure", and "finite-time singularities" as a consequence of divergence or the presence of discontinuity at a finite time, from the viewpoint of dynamical systems.
Moreover, I have also developed several methodologies of rigorous numerics available for various systems possessing the above singular nature, which are usually difficult to treat both mathematically and numerically. The methodologies enable us to compute targeting trajectories with mathematical rigor and concreteness.
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| Academic Significance and Societal Importance of the Research Achievements |
微分方程式の解構造において特徴づけが一般に困難な「特異性」の記述につき、結果の妥当性が限られる数学的結果と数値計算の橋渡しを担う精度保証付き数値計算が、標準的な力学系の道具を用いて適用できるようになる事で、特異性も厳密性と具体性を担保しつつ計算できるようになった。また特異性がいつ生じるかを力学系の言葉で明確にし、その振る舞いを力学系と精度保証付き数値計算で追えるようになった事で、特異性発現の有無を判定する事が可能となり、その振る舞いを厳密性をもって視認する事が可能となった。これは微分方程式の解の特異性解析を容易にかつ体系的にし、信頼性のある深い考察を導く事につながる。
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Report
(4 results)
Research Products
(35 results)
