2023 Fiscal Year Final Research Report
Elucidation of the mechanism of solution singularity formation induced by time lag and mathematical analysis of solution dynamics by such mechanism
| Project/Area Number |
19K21836
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| Research Category |
Grant-in-Aid for Challenging Research (Exploratory)
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| Allocation Type | Multi-year Fund |
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| Review Section |
Medium-sized Section 12:Analysis, applied mathematics, and related fields
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| Research Institution | Shibaura Institute of Technology |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
石渡 恵美子 東京理科大学, 理学部第一部応用数学科, 教授 (30287958)
中田 行彦 青山学院大学, 理工学部, 准教授 (30741061)
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| Project Period (FY) |
2019-06-28 – 2024-03-31
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| Keywords | 遅延微分方程式 / タイムラグ / 解の爆発 / 周期解 / 解のダイナミクス / 安定性 |
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| Outline of Final Research Achievements |
Delay differential equations are one mathematical model that incorporates the effect of time lag.
This study mainly discusses the blow-up problem of nonlinear delay differential equations. Our research has led to a groundbreaking discovery: the phenomenon of the finite-time blow-up of the solution induced by the interaction with the time lag and the nonlinearity of the equations. This finding, accompanied by a rigorous mathematical proof, has profound global implications for understanding nonlinear dynamics. It opens up new avenues of research and application in various fields. Furthermore, we have obtained sufficient conditions for blow-up of solutions for scalar delay differential equations through a comparison with ordinary differential equations. Moreover, we considered the case where the history reference is information about the solution in a specific time interval in the past and obtained sufficient conditions for the solution explosion.
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| Free Research Field |
応用数学
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| Academic Significance and Societal Importance of the Research Achievements |
本研究で得られた遅延誘導爆発現象は、これまで知られていなかった現象であり、それを数学的に明らかにしたことは学術的な価値があり、この成果をベースに非線形遅延微分方程式の研究が活性化することが期待される。また、本研究から得られる時間遅れに関する理解は、ある現象の数理モデル化の過程において、時間遅れを無視できるか取り入れるべきかについての知見を与えることが期待できる。よって、現象の理解やシステムの制御など、関連する科学や工学へも新たな視点を与えることができるため、社会的な意義も高い。
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