平面时变Hamilton系统周期解的存在性和重性

发布时间：2018-06-12 23:44

本文选题：Hamilton系统 + 时变位势　；参考：《苏州大学》2016年博士论文

【摘要】：本文应用Poincaré-Birkhoff扭转定理与拓扑度理论研究平面时变Hamilton系统的周期解的存在性和重性.包括如下三个问题：一、解快速振动的时变Hamilton系统的无穷多周期解的存在性；二、解慢速振动的时变Hamilton系统的无穷多次调和解的存在性；三、由时间映射描述的共振问题.当平面时变Hamilton系统是一个自治系统的扰动时,人们往往可以通过自治系统的能量函数估计扰动系统解的行为,从而进行相平面分析.再应用合适的非线性分析的工具.但如果平面Hamilton系统不是自治系统的扰动(如二阶时变位势方程)时,上述做法不再有效.即便是简单的二阶超线性Hill方程,也会出现解的逃逸,从而系统的Poincaré映射没有定义,给相平面分析带来很大困难.因此,对于此类模型,除掉Jacobowitz和Hartman的经典结果外,其无穷多周期解存在性的结果较少.本文在前两个问题中通过分析解快速振动或解慢速振动的时变Hamilton系统解的盘旋性质(典型的例子是二阶超线性或次线性的时变位势方程和Hill方程,p-超线性或p-次线性的一维p-Laplacian方程),在解盘旋半径估计的基础上,构造解全局存在且在相平面的某个环域上扭转的辅助系统.对辅助系统应用Poincaré-Birkhoff扭转定理得到周期解的存在性,然后利用所得周期解的旋转角度估计回到原方程.这种新的方法基于相平面的几何分析,发展了Jacobowitz和Hartman所用的解析估计的方法.我们的结果把Jacobowitz和Hartman的工作推广到了一维p-Laplacian方程和部分超线性的二阶方程.本文的第三个问题考虑二阶自治方程在共振点处的强迫扰动.通过分析自治系统时间映射的性质来研究强迫方程的周期解的存在性,讨论过程要用到比较精细的相平面分析.所得结果部分回答了Capietto, Mawhin和Zanolin曾经提出的由时间映射方法讨论共振现象的一个问题,推广了他们的相应定理.
[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.
【学位授予单位】：苏州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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