本文關(guān)鍵詞：幾類光滑與非光滑系統(tǒng)的周期解問題　出處：《上海師范大學(xué)》2017年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： Melnikov函數(shù) 分段光滑系統(tǒng) 極限環(huán) 周期解 非自治系統(tǒng)

【摘要】：本文首先對(duì)推廣的Melnikov函數(shù)方法進(jìn)行了補(bǔ)充和完善,建立了分段光滑系統(tǒng)的Melnikov函數(shù)零根與極限環(huán)個(gè)數(shù)之間的關(guān)系.其次,利用推廣的Melnikov函數(shù)方法,研究了兩類分段光滑系統(tǒng).通過復(fù)雜的計(jì)算,求得Melnikov函數(shù)零根的個(gè)數(shù),由此分別得到了這兩類系統(tǒng)由閉軌擾動(dòng)產(chǎn)生的極限環(huán)的個(gè)數(shù).作為對(duì)動(dòng)力系統(tǒng)周期解的另一項(xiàng)研究,本文還考慮了一維周期方程,并給出了幾種方法來研究周期解的個(gè)數(shù)問題和穩(wěn)定性問題.其中,我們通過建立Poincare映射、將方程化成規(guī)范型或利用平均方程,來確定周期解的個(gè)數(shù).另外,通過研究零解的重?cái)?shù),可以了解零解的穩(wěn)定性.本文分為五章,具體安排如下:本文的第一章是緒論,主要介紹了本文的研究背景,包括研究對(duì)象以及所用到的主要方法.本文的第二章介紹了分段光滑系統(tǒng)的Melnikov函數(shù)方法.針對(duì)分段光滑的近哈密頓系統(tǒng),我們對(duì)與推廣的Melnikov函數(shù)方法相關(guān)的結(jié)論進(jìn)行總結(jié)歸納,并且在這些已知結(jié)果的基礎(chǔ)上,給出了周期帶分支定理,以及在Hopf分支問題中的Melnikov函數(shù)的性質(zhì).本文的第三章研究了一類非光滑的Lienard系統(tǒng)的極限環(huán)分支問題.針對(duì)該切換系統(tǒng)的兩種情況:切換直線在x軸上和切換直線在y 軸上,我們先分別討論了帶雙參數(shù)擾動(dòng)的兩類系統(tǒng),并利用帶雙參數(shù)的Melnikov函數(shù)的第一項(xiàng)和第二項(xiàng)表達(dá)式,計(jì)算出這兩類系統(tǒng)的極限環(huán)在閉軌附近的個(gè)數(shù).然后,再根據(jù)這兩類系統(tǒng)與原系統(tǒng)之間的關(guān)系,得出原系統(tǒng)在這兩種情況下的極限環(huán)的個(gè)數(shù),并舉例說明結(jié)論.本文的第四章主要考慮了一類具有Lienard形式的切換系統(tǒng),該系統(tǒng)的左右子系統(tǒng)均為多項(xiàng)式系統(tǒng).通過分段光滑系統(tǒng)的Melnikov函數(shù)方法,我們得到了系統(tǒng)的Melnikov函數(shù)M(h)的表達(dá)式.之后,為了能得到該式的零根個(gè)數(shù),從而確定系統(tǒng)在閉軌附近的極限環(huán)個(gè)數(shù),我們將M(h)中與h有關(guān)的項(xiàng)分為三部分來討論,最終的結(jié)果與這三部分有關(guān).最后,我們按照系統(tǒng)中多項(xiàng)式的次數(shù)大小分類討論,得到一系列關(guān)于這類系統(tǒng)的極限環(huán)在閉軌附近的個(gè)數(shù)的結(jié)果.本文的第五章討論了一維周期系統(tǒng)的周期解的存在性、穩(wěn)定性及其個(gè)數(shù)問題,并給出了研究這些問題的若干理論與方法.本章的主要結(jié)果包含四個(gè)部分:第一部分闡述了周期解個(gè)數(shù)和Poincare映射之間的關(guān)系,并且總結(jié)和改進(jìn)了已有結(jié)論;在第二部分中,我們得到了一維周期方程零解的重?cái)?shù)與穩(wěn)定性之間的關(guān)系,并且分別給出了 x = 0是奇數(shù)重、偶數(shù)重和中心型的條件;第三部分給出了一個(gè)適用于一般方程的規(guī)范型定理;第四部分是平均法理論,闡述了如何利用平均方程來研究周期方程周期解的個(gè)數(shù)問題.
[Abstract]:In this paper, the generalized Melnikov function method is supplemented and perfected, and the relation between the zero root of Melnikov function and the number of limit cycles of piecewise smooth system is established. Two classes of piecewise smooth systems are studied by using the generalized Melnikov function method. The number of zero roots of Melnikov function is obtained by complex calculation. The number of limit cycles generated by the closed-orbit perturbation is obtained respectively. As another study of the periodic solution of the dynamical system, the one-dimensional periodic equation is also considered in this paper. Several methods are given to study the number and stability of periodic solutions. By establishing Poincare maps, the equations are transformed into normal form or mean equations. In addition, by studying the multiplicity of zero solutions, we can understand the stability of zero solutions. This paper is divided into five chapters, the specific arrangements are as follows: the first chapter of this paper is an introduction. This paper mainly introduces the research background. The second chapter introduces the Melnikov function method of piecewise smooth system, aiming at the piecewise smooth near Hamiltonian system. We sum up the conclusions related to the generalized Melnikov function method and give the theorem of periodic band bifurcation on the basis of these known results. In chapter 3, we study the limit cycle bifurcation of a class of nonsmooth Lienard systems. Situation:. Switch the straight line on the x axis and switch the line on the y axis. We first discuss two kinds of systems with two parameters perturbation, and use the first and second expressions of Melnikov function with two parameters. Then, according to the relationship between the two kinds of systems and the original system, the number of limit cycles of the original system in these two cases is obtained. In chapter 4th, we mainly consider a class of switched systems with Lienard form. The left and right subsystems of the system are polynomial systems. By using the Melnikov function method of piecewise smooth systems, we obtain the expression of the Melnikov function of the system. In order to obtain the number of zero roots of the formula and determine the number of limit cycles of the system near the closed orbit, we divide the term of h into three parts to discuss, and the final result is related to these three parts. Finally. We discuss it according to the degree of polynomial in the system. We obtain a series of results about the number of limit cycles near closed orbits of this kind of systems. In chapter 5th, we discuss the existence, stability and number of periodic solutions of one-dimensional periodic systems. The main results of this chapter include four parts: the first part describes the relationship between the number of periodic solutions and Poincare mapping. And summarized and improved the existing conclusions; In the second part, we obtain the relation between the multiplicity and stability of zero solution of one-dimensional periodic equation, and give the condition that x = 0 is odd, even and central. In the third part, a normal form theorem for general equations is given. The 4th part is the theory of the averaging method. How to use the average equation to study the number of periodic solutions of the periodic equation is discussed.
【學(xué)位授予單位】：上海師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

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