本文選題：仿射周期解 + 周期解�。� 參考：《吉林大學(xué)》2017年博士論文

【摘要】：微分方程的擾動理論被科學(xué)家們所重點關(guān)注始于十八世紀.人們利用牛頓的萬有引力理論來研究行星與太陽所構(gòu)成兩體問題時,發(fā)現(xiàn)所得到的的結(jié)果與實際觀測的情況并不是很吻合,為了解釋這種現(xiàn)象人們猜想這可能是由于行星的運動除了受太陽影響以外還受衛(wèi)星和其它大行星等因素的影響.這種想法導(dǎo)致了擾動兩體問題的誕生.自從Laplace[40]和Lagrange[39]建立了平均原理開始,其一直都是研究周期擾動系統(tǒng)的有力工具.然而很多現(xiàn)象除了具有時間上的周期性以外還具有空間上的對稱性,本文我們將研究一類同時具有時間周期和空間對稱的系統(tǒng),我們稱之為仿射周期系統(tǒng).對于一個周期的微分方程來說,一個很自然的問題就是尋找周期解,與之相對應(yīng)的,本篇博士論文所關(guān)注的問題就是在仿射周期系統(tǒng)中是否存在著具有同樣對稱結(jié)構(gòu)的解,我們稱之為仿射周期解.在第一章中我們簡單的介紹了擾動系統(tǒng)和平均原理的起源與發(fā)展進程,并且給出了近些年來利用高階平均原理來尋找周期解的結(jié)果.我們介紹了 Mawhin的重合度理論和Krasnosel'skii與Perov使用拓撲度理論所得到的一個關(guān)于周期解存在性的有趣的定理,人們稱之為Krasnosel'skii-Perov存在性定理.在第一章的最后我們給出了仿射周期系統(tǒng)的定義,以及其相關(guān)的一些工作,同時也介紹了本文的主要結(jié)果.在第二章中我們建立了擾動仿射周期系統(tǒng)的平均原理.對于一個擾動的周期系統(tǒng),如果其向量場的平均函數(shù)在小參數(shù)為零時具有非退化的零點,則可以知道其有周期解,這就是經(jīng)典的一階平均原理.對于擾動的仿射周期系統(tǒng)而言,我們發(fā)現(xiàn)其周期解的存在性與其仿射矩陣的性質(zhì)有關(guān).如果單位矩陣與仿射矩陣的差是可逆的,當(dāng)小參數(shù)足夠小的時候,系統(tǒng)自然存在仿射周期解.由于周期系統(tǒng)的仿射矩陣就是單位矩陣,所以這個性質(zhì)在周期情形下是體現(xiàn)不出來的.當(dāng)單位矩陣與仿射矩陣的差不可逆的時候,我們分別建立了一階的平均原理和高階的平均原理.我們發(fā)現(xiàn)如果其一階擾動向量場的平均函數(shù)在單位矩陣與仿射矩陣差的核空間上的投影函數(shù)滿足某種拓撲度不為零的性質(zhì),同樣可以得到仿射周期解的存在性.這可以看成是經(jīng)典周期系統(tǒng)的平均原理的自然推廣,因為在仿射矩陣等于單位矩陣的時候,這個結(jié)果與周期系統(tǒng)的一階平均原理是一致的.對于高階的擾動系統(tǒng),相比于一階情形,除了要求其各階擾動向量場的平均函數(shù)的和函數(shù)的拓撲度不為零以外,我們還需要其擾動函數(shù)滿足一些額外的性質(zhì).這是因為我們使用的方法主要是基于Mawhin的重合度理論,要利用到拓撲度的同倫不變性,需要同倫映射在邊界上的取值不為零.根據(jù)其要求的性質(zhì)不同,我們得到了兩個不同的結(jié)果.與已有的結(jié)果相比,即使在仿射矩陣等于單位矩陣也就是周期情形下,我們的結(jié)果也完全是新的.近些年來對于周期系統(tǒng)的高階平均原理,主要是使用Poincare的方法得到的.通過建立Poincare映射,再把其按小參數(shù)做Taylor展開,利用擾動函數(shù)拓撲度的性質(zhì)得到Poincare映射的不動點,從而得到系統(tǒng)的周期解.與其相比,我們需要向量場滿足一些額外的條件,然而我們對系統(tǒng)的光滑性要求更低,而且我們所要用到的平均函數(shù)恰好就是向量場各階擾動平均函數(shù)的和,計算起來相對容易,而已有的方法由于要把解按小參數(shù)展開,需要依次計算各階變分,當(dāng)階數(shù)高的時候這個計算一般是比較繁瑣的.在第三章中我們建立了仿射周期系統(tǒng)的Krasnosel'skii-Perov型存在性定理.對于一個有界區(qū)域上的周期系統(tǒng),Krasnosel'skii和Perov[35,36]在上個世紀五六十年代使用拓撲度的方法證明了如果系統(tǒng)從邊界上出發(fā)的解在小于等于一個周期的時間內(nèi)都不會回到出發(fā)點,并且向量場在零時刻的取值函數(shù)的拓撲度不為零,系統(tǒng)就會存在一個周期解.我們分別從兩個方面將這個結(jié)果推廣到了仿射周期系統(tǒng)上.一是利用重合度的思想,將其約化到單位矩陣與仿射矩陣差的核空間上.我們證明了如果從在核空間的投影是在邊界上的點出發(fā)的解,在小于等于一個周期的時間內(nèi)通過一個仿射變換不會回到出發(fā)點,而經(jīng)過一個仿射變換能在小于等于一個周期的時間內(nèi)回到出發(fā)點的解都不會達到區(qū)域的邊界上,再加上向量場零時刻在核空間上的投影函數(shù)的拓撲度不為零,那么系統(tǒng)就會有仿射周期解.第二個結(jié)果我們是在全空間上考慮,我們證明了如果能找到一個連續(xù)的矩陣函數(shù),使得其在T時刻的取值恰好就是仿射矩陣,而在零時刻的取值與單位矩陣的差是可逆的,如果系統(tǒng)從邊界上出發(fā)的點在小于等于一個周期的時間內(nèi)經(jīng)過相同時刻矩陣函數(shù)的變換不會回到出發(fā)點,那么系統(tǒng)就會存在仿射周期解.由于Krasnosel'skii-Perov存在性定理的條件在實際使用中是比較難驗證的,在第三章的第二節(jié)針對擾動系統(tǒng)我們給出了一個條件相對容易驗證的結(jié)果.假設(shè)向量場滿足某些性質(zhì),并且仍然要求系統(tǒng)在零時刻取值函數(shù)的拓撲度不為零,同樣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.
【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O175

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