A billiard problem in nonlinear and nonequilibrium systems: bifurcation with symmetry and degeneracy
| Project/Area Number |
16K17649
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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 | Meiji University |
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Principal Investigator |
Miyaji Tomoyuki 明治大学, 研究・知財戦略機構, 特任准教授 (20613342)
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
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| Keywords | 力学系の分岐 / ビリヤード問題 / 自己駆動粒子 / 反応拡散系 / 進行スポット / 非線形・非平衡系 / 力学系 / 分岐 / 数値解析 / 縮約方程式 |
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| Outline of Final Research Achievements |
Orbits of a self-propelling particle in nonlinear and nonequilibrium systems are studied from viewpoint of mathematics. Such particles are observed in many systems. In contrast to classical billiard problems, the dynamical system of the particle has an attractor. As the aspect ratio of a rectangular domain varies, the attractor becomes periodic or nonperiodic intermittently. In this study, it turned out that such an intermittent change can be understood as a kind of synchronization between vertical and horizontal oscillatory modes and that its origin is a symmetry breaking bifurcation point of the rest state at which stable rest state loses stability and stationary motion occurs.
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| Academic Significance and Societal Importance of the Research Achievements |
水面を動く樟脳円板や油滴，非線形光学の数理モデルなど自己駆動粒子系は科学の様々な系で観察され，研究されている．平面上の反応拡散系における進行スポット解が典型例であるが，それらの系で自己駆動粒子は漸近的な等速直線運動と領域境界での非完全弾性反射を繰り返すようである．自己駆動粒子研究の多くは多数の自己駆動粒子間の相互作用から生じる集団的運動に注目している．本研究では単一の粒子が領域境界と相互作用することで描かれる軌道を研究するものである点が新しい．縮約された常微分方程式系を分岐理論によって研究することで，自己駆動力を生み出す機構の詳細に依存しない普遍的な性質を数学的に示しうることは意義深い．
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Report
(4 results)
Research Products
(12 results)
