Several Classes Of Bifurcations Of Limit Cycles And Isochronous Centers For Differential Autonomous Systems

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.