Generalized Center, Isochronous Center And Bifurcation Of Limit Cycles For Several Classes Of Planar Differential Systems

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.