2012 Fiscal Year Final Research Report
New construction of phase space analysis that conforms to low-dimensional dynamical systems
| Project/Area Number |
22540190
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Shimane University |
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Principal Investigator |
SUGIE Jitsuro 島根大学, 総合理工学研究科, 教授 (40196720)
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| Co-Investigator(Renkei-kenkyūsha) |
MATSUNAGA Hideaki 大阪府立大学, 工学研究科, 准教授 (40332960)
SAITO Yasuhisa 島根大学, 総合理工学研究科, 准教授 (30402241)
YAMAOKA Naoto 大阪府立大学, 工学研究科, 准教授 (90433789)
ONITSUKA Nasakazu 岡山理科大学, 理学部, 講師 (20548367)
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| Project Period (FY) |
2010 – 2012
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| Keywords | 関数方程式論 / 相平面解析 / 漸近安定性 / 半分線形系 / 優線形振動子 / 概周期関数 / 船舶の自由横揺れ / 極限閉軌道 |
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| Research Abstract |
Phase space (or plane) analysis was improved and constructed in order to clarify criteria which judge whether the equilibrium (or the zero solution) of low- dimensional linear differential systems, half-linear differential systems, or quasi-linear differential systems is asymptotically stable, globally asymptotically stable, or uniformly globally asymptotically stable.The obtained criteria were applied to the analysis of the free rolling motion of a small ship. It is well- known that the resistance of free rolling motion of a small fishing vessel is mainly classified into three types: frictional resistance; eddy-making resistance (or viscous pressure resistance); wave-making resistance. Although wave-making resistance is proportional to the angular velocity, frictional resistance and eddy-making resistance are proportional to the square of the angular velocity. For this reason, the equation of rolling motion need be described by a damped super-linear oscillator. In this research, a necessary and sufficient condition for the equilibrium of the damped super-linear oscillator to be globally asymptotically stable was found out.The above-mentioned criteria were also applied to a Lotka-Volterra predator-prey model with prey (or predator) receiving time-variation of the environment. This model is very famous as an ecology model and have a unique interior equilibrium. In this research, sufficient conditions for the interior equilibrium to be globally asymptotically stable were obtained. Moreover, the uniqueness problem of limit cycles about a Rosenzweig MacArthur predator-prey model in consideration of immigration of prey was discussed.
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