【作者】 杜超雄；

【导师】 刘一戎；

【作者基本信息】 中南大学 ， 应用数学， 2009， 博士

【摘要】 本篇博士论文以计算机代数系统为工具对几类平面微分系统的极限环分枝、中心与广义中心问题、等时中心问题进行了一些研究,同时将已有理论应用到生态系统中去研究一些生态模型的定性性质.全文共分七章.第一章对平面多项式微分系统的极限环分枝、中心与等时中心以及定性与分枝理论在生态系统中的应用等问题的历史背景和研究现状进行了综述,并归纳了本文的特色工作.第二章给出了一种寻找Zn等变系统的简便方法,并对Zn等变向量场中的微分系统的一些性质做了一些归纳总结.同时,以一类Z8-等变对称的七次系统为研究示例,在个人计算机上推导出八个拓扑结构相同的焦点中其中一个的前5个奇点量,进而得出其前5阶焦点量,并得出由八个拓扑结构相同的焦点共可在一定条件下分支出40个极限环的好的实例,同时找出了它的分枝条件及极限环稳定性的判断条件.第三章主要是关于三次系统等变对称结构的分析和几类具有等变对称性的三次系统的极限环分支的研究,一种寻找等变对称系统的方法被给出.并对未曾研究过的具有不同等变对称结构的三次系统（即z。等变对称的三次系统、Z₃,等变对称的三次系统、Z₄等变对称的三次系统）的定性与分支行为进行了一一的研究.本章的工作是现有研究结论的有益的补充.（这一结论已经提交给《湘潭大学学报》）第四章主要研究了一类广义对称的平面九次微分自治系统的中心焦点问题与极限环分枝行为.通过作Bendixson倒径变换与时间变换,以及广义焦点量的仔细计算,我们得出了该系统的无穷远点与三个初等奇点能够同时成为广义中心的条件,并进行了严格的证明.同时我们得出在一定条件下,这四个广义焦点能够分枝出20个极限环的结果,其中包括5个大振幅极限环和15个小振幅极限环.这种结论到目前为止是新的.第五章研究一类广义等变对称的九次系统,通过周期常数（或称为等时常数）的计算与分析,找出了无穷远点与三个初等奇点同时成为等时中心（或称为可线性化中心）的必要条件,进一步证明了这些必要条件也是充分条件.无穷远点与几个初等奇点同时成为等时中心的结论是第一次给出,在以往的文献中未有发现,我们的结论有一定意义.第六章研究一类拟对称七次系统的广义中心、等时中心与极限环分枝问题,得出了该系统的无穷远点与初等奇点同时成为广义中心与等时中心的条件并进行了严格的证明,进一步给出该系统可以由无穷远点分枝出5个赤道环的同时还可以由初等奇点分枝出5个小振幅极限环.这种结论是罕见的,本章的结论是新的.第七章利用微分方程定性与分支理论研究了一类食饵具有常投放的稀疏效应捕食系统,得到了存在唯一极限环和不存在极限环及系统全局渐近稳定的充要条件.此外,还研究了一类三次Kolmogorov捕食系统,算出了其正平衡点（1,1）的前五阶焦点量,同时找到了5个极限环（或少于5个极限环）的分枝条件与这些极限环的相对位置.特别地,文中还显示该系统可以有3个稳定的极限环,就三次Kolmogorov系统的稳定环的数目而言,这一结论是最好的.更多还原

【Abstract】 This Ph.D. Thesis is devoted to study the bifurcations of limit cycle and the problem of center and ischoronous center for several classes of planar differential systems with the help of computer’s algebra system, at the same time, the qualitative theory and bifurcation theory are applied to investigate the qualitative nature of two class of ecological systems or ecological models. It is composed of seven chapters.In Chapter 1, we introduce the historical background and the present progress of problems which are concerned with centers, isochronous centers, and bifurcations of limit cycles for planar polynomial differential systems. And the qualitative nature of ecological model is considered. The main works of this paper are concluded as well.In Chapter 2, a kind of simple method to find Z_n -equivariant systems is given, the quality of differential systems in Z_n-equivariant vector field is summarized. At the same time,as an example ,a class of seventh degree Z₈-equivariant systems is investigated, and the first five focal values are given. Moreover, it is shown that 40 limit cycles can bifurcate from this class of system. The conditions of bifurcation and stability are obtained.In Chapter 3, our work is concerned with the analysis of cubic system’s equivariant symmetric structure and the investigation of limit cycles bifurcation for several classes of equivariant cubic systems. A kind of method to find equivariant symmetric systems is given, which is significative for researching the bifurcation behave -or of Z_n -equivariant systems.In terms of equivariant symmetric cubic system, we investigate other three cases （i.e., Z_∞-equivariant case , Z₃ -equivariant case and Z₄ -equivariant case） except studied Z₂ -equivariant case and obtain center condition and bifurcation results of each case .These results are supplements for some studied references about cubic system.In Chapter 4, our researches are concerned with the center-focus problem and the limit cycle bifurcation problem for a class of planar general equivariant system of nine degrees. By making Bendixson transformation and time transformation of system and calculating general focal values carefully, we obtain the conditions that the infinity and three elementary focuses become four general centers at the same time. More -over, 20 limit cycles including 15 small limit cycles from three element -ary foci and 5 large limit cycles from the infinity can occur under a certain condition. What is worth mentioning is that similar conclusions are hardly seen in published papers up till now and our work is completely new.In Chapter 5, a class of general equivariant system of nine degrees is investigated. Through the calculation and the analysis of periodic constants （or called ischoronous constants ）, we obtain the conditions that the infinity and three elementary singular points become isochronous centers at the same time, and these conditions are also proved to be sufficient conditions of isochronicity. It is hardly seen in published articles for the result that the the infinity and elementary singular points become isochronous centers at the same time, our work is significant.In Chapter 6, general center and isochronous centers and limit cycles bifurcation for a class of quasi-symmetric seventh degree system are considered. we obtain the conditions that the infinity and the elementary singular point become isochronous centers at the same time, and prove them strictly. Moreover, we give the conclusions that the infinity can bifurcate 5 large limit cycles and the elementary focus can bifurcate 5 small limit cycles under the same condition. This kind of work is new.In Chapter 7, firstly, a class of predator-prey system with scanty effect of which the prey species possesses a constant invest is studied by making use of qualitative and bifurcation theory, and we obtain sufficient and necessary condition for the existence and uniqueness of limit cycle surrounding the positive equilibrium point and for the global stability of the system. Secondly, A class of cubic Kolmogorov system are considered and we obtain 5 focal values of the positive equilibrium point （1,1）,at the same time the condition for the existence of 5 limit cycles or less surrounding the positive equilibrium point （1,1） and the locality of 5 limit cycles is given.Especially,3 stable limit cycles are shown in this paper ,which is the best result in terms of the number of stable limit cycles for cubic Kolmogorov system.更多还原