扰动系统的仿射周期解

发布时间：2018-06-19 18:16

本文选题：仿射周期解 + 周期解　；参考：《吉林大学》2017年博士论文

【摘要】：微分方程的扰动理论被科学家们所重点关注始于十八世纪.人们利用牛顿的万有引力理论来研究行星与太阳所构成两体问题时,发现所得到的的结果与实际观测的情况并不是很吻合,为了解释这种现象人们猜想这可能是由于行星的运动除了受太阳影响以外还受卫星和其它大行星等因素的影响.这种想法导致了扰动两体问题的诞生.自从Laplace[40]和Lagrange[39]建立了平均原理开始,其一直都是研究周期扰动系统的有力工具.然而很多现象除了具有时间上的周期性以外还具有空间上的对称性,本文我们将研究一类同时具有时间周期和空间对称的系统,我们称之为仿射周期系统.对于一个周期的微分方程来说,一个很自然的问题就是寻找周期解,与之相对应的,本篇博士论文所关注的问题就是在仿射周期系统中是否存在着具有同样对称结构的解,我们称之为仿射周期解.在第一章中我们简单的介绍了扰动系统和平均原理的起源与发展进程,并且给出了近些年来利用高阶平均原理来寻找周期解的结果.我们介绍了 Mawhin的重合度理论和Krasnosel'skii与Perov使用拓扑度理论所得到的一个关于周期解存在性的有趣的定理,人们称之为Krasnosel'skii-Perov存在性定理.在第一章的最后我们给出了仿射周期系统的定义,以及其相关的一些工作,同时也介绍了本文的主要结果.在第二章中我们建立了扰动仿射周期系统的平均原理.对于一个扰动的周期系统,如果其向量场的平均函数在小参数为零时具有非退化的零点,则可以知道其有周期解,这就是经典的一阶平均原理.对于扰动的仿射周期系统而言,我们发现其周期解的存在性与其仿射矩阵的性质有关.如果单位矩阵与仿射矩阵的差是可逆的,当小参数足够小的时候,系统自然存在仿射周期解.由于周期系统的仿射矩阵就是单位矩阵,所以这个性质在周期情形下是体现不出来的.当单位矩阵与仿射矩阵的差不可逆的时候,我们分别建立了一阶的平均原理和高阶的平均原理.我们发现如果其一阶扰动向量场的平均函数在单位矩阵与仿射矩阵差的核空间上的投影函数满足某种拓扑度不为零的性质,同样可以得到仿射周期解的存在性.这可以看成是经典周期系统的平均原理的自然推广,因为在仿射矩阵等于单位矩阵的时候,这个结果与周期系统的一阶平均原理是一致的.对于高阶的扰动系统,相比于一阶情形,除了要求其各阶扰动向量场的平均函数的和函数的拓扑度不为零以外,我们还需要其扰动函数满足一些额外的性质.这是因为我们使用的方法主要是基于Mawhin的重合度理论,要利用到拓扑度的同伦不变性,需要同伦映射在边界上的取值不为零.根据其要求的性质不同,我们得到了两个不同的结果.与已有的结果相比,即使在仿射矩阵等于单位矩阵也就是周期情形下,我们的结果也完全是新的.近些年来对于周期系统的高阶平均原理,主要是使用Poincare的方法得到的.通过建立Poincare映射,再把其按小参数做Taylor展开,利用扰动函数拓扑度的性质得到Poincare映射的不动点,从而得到系统的周期解.与其相比,我们需要向量场满足一些额外的条件,然而我们对系统的光滑性要求更低,而且我们所要用到的平均函数恰好就是向量场各阶扰动平均函数的和,计算起来相对容易,而已有的方法由于要把解按小参数展开,需要依次计算各阶变分,当阶数高的时候这个计算一般是比较繁琐的.在第三章中我们建立了仿射周期系统的Krasnosel'skii-Perov型存在性定理.对于一个有界区域上的周期系统,Krasnosel'skii和Perov[35,36]在上个世纪五六十年代使用拓扑度的方法证明了如果系统从边界上出发的解在小于等于一个周期的时间内都不会回到出发点,并且向量场在零时刻的取值函数的拓扑度不为零,系统就会存在一个周期解.我们分别从两个方面将这个结果推广到了仿射周期系统上.一是利用重合度的思想,将其约化到单位矩阵与仿射矩阵差的核空间上.我们证明了如果从在核空间的投影是在边界上的点出发的解,在小于等于一个周期的时间内通过一个仿射变换不会回到出发点,而经过一个仿射变换能在小于等于一个周期的时间内回到出发点的解都不会达到区域的边界上,再加上向量场零时刻在核空间上的投影函数的拓扑度不为零,那么系统就会有仿射周期解.第二个结果我们是在全空间上考虑,我们证明了如果能找到一个连续的矩阵函数,使得其在T时刻的取值恰好就是仿射矩阵,而在零时刻的取值与单位矩阵的差是可逆的,如果系统从边界上出发的点在小于等于一个周期的时间内经过相同时刻矩阵函数的变换不会回到出发点,那么系统就会存在仿射周期解.由于Krasnosel'skii-Perov存在性定理的条件在实际使用中是比较难验证的,在第三章的第二节针对扰动系统我们给出了一个条件相对容易验证的结果.假设向量场满足某些性质,并且仍然要求系统在零时刻取值函数的拓扑度不为零,同样可以得到仿射周期解的存在性。
[Abstract]:The perturbation theory of differential equations has been the focus of scientists' attention in eighteenth Century. People use Newton's theory of universal gravitation to study the two body problem of planets and the sun, and find that the results are not very consistent with the actual observations. In order to explain this phenomenon, it is assumed that this may be due to the planets. In addition to the influence of the sun, the movement is influenced by the satellite and other planets. This idea leads to the birth of the two body problem. Since Laplace[40] and Lagrange[39] have established the mean principle, it has always been a powerful tool to study the periodic disturbance system. However, many phenomena have the periodicity of time. In addition to the symmetry in space, we will study a class of systems with both time and space symmetry. We call it an affine periodic system. For a periodic differential equation, a very natural problem is to find periodic solutions. We call it an affine periodic solution in the affine periodic system. In the first chapter, we briefly introduce the origin and development process of the perturbation system and the mean principle, and give the results of finding the periodic solution by the high order mean principle in recent years. We introduce the coincidence of the Mawhin. Degree theory and an interesting theorem on the existence of periodic solutions by Krasnosel'skii and Perov, which are called Krasnosel'skii-Perov existence theorems. In the end of Chapter 1, we give the definition of the affine periodic system, and some related work, and also introduce the main conclusion of this paper. In the second chapter, we have established the mean principle of the perturbed affine periodic system. For a periodic system of a disturbance, if the average function of its vector field has a non degenerate zero point when the small parameter is zero, we can know that it has a periodic solution. This is the classical first order mean principle. For the affine periodic system of disturbance, I am concerned. We find that the existence of the periodic solution is related to the properties of the affine matrix. If the difference between the unit matrix and the affine matrix is reversible, when the small parameter is small enough, the system naturally has an affine periodic solution. Because the affine matrix of the periodic system is a unit matrix, this property is not reflected in the periodic case. When the difference of the difference between the unit matrix and the affine matrix is irreversible, we establish the first order mean principle and the higher order mean principle respectively. We find that if the average function of the first order perturbation field of the first order perturbation field in the kernel space of the unit matrix and the affine matrix difference satisfies the property that some topological degree is not zero, we can get the same. The existence of the affine periodic solution, which can be regarded as the natural generalization of the mean principle of the classical periodic system, because when the affine matrix equals the unit matrix, the result is consistent with the first order mean principle of the periodic system. The topological degree of the sum function of the mean function is not zero, and we also need its perturbation function to satisfy some additional properties. This is because the method we use is mainly based on the coincidence degree theory of Mawhin, to make use of the homotopy invariance of the topological degree, and the value of the homotopy mapping on the boundary is not zero. We get two different results. Compared with the existing results, our results are completely new even if the affine matrix is equal to the unit matrix or the periodic case. In recent years, the high order mean principle of the periodic system is mainly obtained by using the Poincare method. By establishing the Poincare mapping, then it is smaller. The parameter is expanded by Taylor, and the fixed point of the Poincare mapping is obtained by the property of the topological degree of the disturbance function, thus the periodic solution of the system is obtained. The sum of the dynamic average functions is relatively easy to calculate, and the existing methods need to calculate each order variation in turn because of the small parameters to be expanded. When the order is high, this calculation is generally more complicated. In the third chapter, we establish the Krasnosel'skii-Perov existence theorem of the affine periodic system. The periodic system on the domain, Krasnosel'skii and Perov[35,36] use the method of topological degree in the 50s and 60s of last century to prove that if the solution from the boundary will not return to the starting point in a time less than one period, and the topological degree of the vector field at zero time is not zero, the system will have one. We generalize this result from two aspects to the affine periodic system. First, we use the idea of coincidence to reduce it to the nuclear space of the difference between the unit matrix and the affine matrix. We prove that if the projection from the kernel space is the point on the boundary, it is less than a period of time. A affine transformation does not return to the starting point, and the solution that can return to the starting point after an affine transformation can return to the starting point within a period of less than one period will not reach the boundary of the region. Then the system will have an affine periodic solution. Then the system will have an affine periodic solution. Then the system will have an affine periodic solution. Second knots will be found. We consider it in the whole space. We prove that if we can find a continuous matrix function, the value of the value at T time is just an affine matrix, and the difference between the zero time value and the unit matrix is reversible, if the point on the boundary is reduced to the same moment in the time of the same period. The transformation of the array function will not return to the starting point, then the system will have an affine periodic solution. The condition of the Krasnosel'skii-Perov existence theorem is more difficult to verify in practical use. In the second section of the third chapter, we give a relatively easy verifiable result for the perturbation system. Qualitative, and still require that the topological degree of the value function of the system at zero time is not zero, and the existence of affine periodic solutions can also be obtained.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

【相似文献】