【作者】 黄文韬；

【导师】 刘一戎；

【作者基本信息】 中南大学 ， 概率论与数理统计， 2004， 博士

【摘要】 本篇博士论文主要研究平面微分自治系统中心、等时中心与极限环分支问题，由7章组成。 第一章，对平面多项式微分系统中心、等时中心与极限环分支等问题的历史背景和研究现状进行了综述．归纳了本文所做的工作。 第二章研究了一类三次多项式系统由细焦点分支出极限环的问题，证明了该系统有12个小振幅极限环，证明过程是符号和代数的，且焦点量的表达式相对简单。 在第三章中，我们分别研究了一类三次多项式系统、一类五次多项式系统、一类七次多项式无穷远点的中心条件与极限环分支问题。得到了三次、五次、七次系统分别可以在无穷远点分支出7个、8个、9个大振幅极限环的结论，这几个结果是目前最好的。同时还研究了一类五次系统原点的中心条件及在同步扰动下原点与无穷远点的极限环分支问题。 在第四章，研究了一类五次多项式系统高次奇点与无穷远点的可积性条件与极限环分支问题，这是首次在一个系统中考虑高次奇点与无穷远点同步扰动分支出极限环问题，得到了该系统可以在原点(高次奇点)分支出5个极限环同时在无穷远点分支出2个极限环的结论。 在第五章，给出了计算多项式微分系统周期常数的一种算法，算法是线性递推的，用此算法求周期常数时，只需以系统右端系数为符号进行有限次加、减、乘、除四则运算，避免了现有方法的复杂的积分与三角函数运算，而且在计算机代数系统中用一个递归函数就能方便的实现。给出了等时中心的一个新的充分条件，作为递推公式的应用，我们还解决了一类多项式系统原点的中心与等时中心条件问题。 在第六章，给出一种研究一类微分系统无穷远点中心、等时中心与极限环分支的间接方法，通过一个同胚变换把系统无穷远点化为原点进行研究．作为新方法的应用，我们解决了一类多项式系统无穷远点的中心、极限环分支问题，同时还解决了两类有理系统无穷远点的等时中心问题。 最后一章给出了研究一类多项式系统高次奇点性质(可积性条件、极限环分支等)的一种新方法．作为应用，我们在复域中详细地求出了一类五次多项式系统的原点(高次奇点)成为中心、拟等时中心的条件。 每一章的主要计算过程均在附录中给出。更多还原

【Abstract】 This Ph.D.Thesis is devoted to center conditions, isochronous center conditions and bifurcations of limit cycles for planar differential systems. It is composed of seven chapters.In Chapter 1. we introduce the historical background and the present progress of problems that concern with centers, isochronous centers and bifurcations of limit cycles for planar polynomial differential systems. The main works of this paper are concluded as well.In Chapter 2. the problem of limit cycles bifurcating from fine foci for a cubic polynomial system is investigated. We prove that the system has twelve small amplitude limit cycles. The proof of existence of limit cycles is algebraic and symbolic.In Chapter 3. we study the center conditions and the bifurcations of limit cycles of infinity for a cubic polynomial system, a quintic polynomial system and a seven degree polynomial system orderly and. obtain that the cubic polynomial system has 7 limit cycles, the quintic polynomial system has 8 limit cycles and the seven degree polynomial system has 9 limit cycles around infinity respectively. At the same time, the center conditions and bifurcation of limit cycles at the origin of the quintic polynomial system are also investigated.In Chapter 4. we study the center conditions and the bifurcation of limit cycles at a degenerate singular point as well as that at infinity for a quintic polynomial system and prove that the system has 5 limit cycles around the origin (the degenerate singular point) and 2 limit cycles around infinity. This is a first time that the problem of limit cycles bifurcating from a degenerate singular point and from infinity under the synchronous perturbed conditions is investigated.In Chapter 5. We give an algorithm to compute complex period constant. The algorithm is linear recursive and easy to realized by a recursive function in computer algebra systems. With forcing only addition, subtraction, multiplication and division to the coefficients of the system, the period constants can be deduced. Compared with the known methods, complex integrating calculations and operations of trigonometric functions are avoided in computation. We also introduce a new sufficient condition of the isochronous center. As an application of the new algorithm, we study the conditions of the origin to be a center and to be an isochronous center for a class of polynomial system.Chapter 6 gives an indirect method to investigate center conditions, isochronous center conditions and bifurcations of limit cycles at infinity for differential systems. By a homeomorphous transformation, infinity can be transferred the origin and as a result, the properties of infinity can be studied with the methods of the origin. As an application of our method, we solve the problems of center conditions and bifurcation of limit cycles at infinity for a quintic polynomial system. The conditions of infinity to be an isochronous center for two rational systems are also derived respective!}’.At last Chapter, we introduce a new method to investigate the properties, such as integrability and bifurcation of limit cycles for a degenerate singular point of polynomial systems. As an application, we discuss the conditions of the origin (a degenerate singular point) to be a center and to be a quasi-isochronous center in the complex number field for a class of quintic polynomial system.更多还原