本文選題：Hamilton系統(tǒng) + 時變位勢��； 參考：《蘇州大學》2016年博士論文

【摘要】：本文應用Poincaré-Birkhoff扭轉(zhuǎn)定理與拓撲度理論研究平面時變Hamilton系統(tǒng)的周期解的存在性和重性.包括如下三個問題：一、解快速振動的時變Hamilton系統(tǒng)的無窮多周期解的存在性；二、解慢速振動的時變Hamilton系統(tǒng)的無窮多次調(diào)和解的存在性；三、由時間映射描述的共振問題.當平面時變Hamilton系統(tǒng)是一個自治系統(tǒng)的擾動時,人們往往可以通過自治系統(tǒng)的能量函數(shù)估計擾動系統(tǒng)解的行為,從而進行相平面分析.再應用合適的非線性分析的工具.但如果平面Hamilton系統(tǒng)不是自治系統(tǒng)的擾動(如二階時變位勢方程)時,上述做法不再有效.即便是簡單的二階超線性Hill方程,也會出現(xiàn)解的逃逸,從而系統(tǒng)的Poincaré映射沒有定義,給相平面分析帶來很大困難.因此,對于此類模型,除掉Jacobowitz和Hartman的經(jīng)典結(jié)果外,其無窮多周期解存在性的結(jié)果較少.本文在前兩個問題中通過分析解快速振動或解慢速振動的時變Hamilton系統(tǒng)解的盤旋性質(zhì)(典型的例子是二階超線性或次線性的時變位勢方程和Hill方程,p-超線性或p-次線性的一維p-Laplacian方程),在解盤旋半徑估計的基礎(chǔ)上,構(gòu)造解全局存在且在相平面的某個環(huán)域上扭轉(zhuǎn)的輔助系統(tǒng).對輔助系統(tǒng)應用Poincaré-Birkhoff扭轉(zhuǎn)定理得到周期解的存在性,然后利用所得周期解的旋轉(zhuǎn)角度估計回到原方程.這種新的方法基于相平面的幾何分析,發(fā)展了Jacobowitz和Hartman所用的解析估計的方法.我們的結(jié)果把Jacobowitz和Hartman的工作推廣到了一維p-Laplacian方程和部分超線性的二階方程.本文的第三個問題考慮二階自治方程在共振點處的強迫擾動.通過分析自治系統(tǒng)時間映射的性質(zhì)來研究強迫方程的周期解的存在性,討論過程要用到比較精細的相平面分析.所得結(jié)果部分回答了Capietto, Mawhin和Zanolin曾經(jīng)提出的由時間映射方法討論共振現(xiàn)象的一個問題,推廣了他們的相應定理.
[Abstract]:In this paper, we apply Poincar 茅 -Birkhoff 's torsion theorem and topological degree theory to study the existence and nature of periodic solutions for planar time-varying Hamiltonian systems. It includes the following three problems: first, the existence of infinite periodic solutions for time-varying Hamiltonian systems with fast vibration; second, the existence of infinite multiple harmonic solutions for time-varying Hamiltonian systems with slow vibration; and third, the resonance problem described by time maps. When the planar time-varying Hamiltonian system is a disturbance of an autonomous system, the behavior of the solution of the disturbance system can be estimated by the energy function of the autonomous system, and the phase plane analysis can be carried out. Then the appropriate nonlinear analysis tools are applied. However, if the planar Hamiltonian system is not a disturbance of the autonomous system (such as the second order time-varying potential equation), the above method is no longer effective. Even a simple second-order superlinear Hill equation will escape the solution, thus the Poincar 茅 map of the system is not defined, which makes the phase plane analysis very difficult. Therefore, except for the classical results of Jacobowitz and Hartman, the existence of infinite periodic solutions for this kind of model is less than that of Jacobowitz's and Hartman's. In this paper, we analyze the hovering properties of time-varying Hamiltonian systems with fast or slow oscillation solutions in the first two problems (typical examples are the second order superlinear or sublinear time-varying potential equations and Hill equation / p-superlinear or p-sublinear equations). The linear one-dimensional p-Laplacian equation is based on the estimation of the hovering radius of the solution. An auxiliary system with a global existence and torsion on a circular domain of the phase plane is constructed. The existence of periodic solution is obtained by applying Poincar 茅 -Birkhoff 's torsion theorem to the auxiliary system, and then the rotation angle of the obtained periodic solution is estimated back to the original equation. This new method is based on the geometric analysis of the phase plane and develops the analytical estimation method used by Jacobowitz and Hartman. Our results extend the work of Jacobowitz and Hartman to one-dimensional p-Laplacian equations and partially superlinear second-order equations. The third problem in this paper considers the forced perturbation of the second order autonomous equation at the common vibration point. By analyzing the properties of time mapping of autonomous systems, the existence of periodic solutions of forced equations is studied. The results partly answer a problem that has been put forward by Capietto Mawhin and Zanolin to discuss resonance phenomenon by time mapping method and generalize their corresponding theorem.
【學位授予單位】：蘇州大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O175

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相關(guān)期刊論文 前2條

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本文編號：2011584