Hopf Bifurcations And Integrability For Classes Of Planar Differential Systems

This Ph.D. thesis is devoted to the problems of center-focus,isochronicity and bifurcation of limit cycles for planar polynomial autonomous differential systems. It is composed of seven chapters.In Chapter1, the historical background and the present progress of problems con-cerned with centers, integrability, isochronous centers, linearizability and bifurcation of limit cycles for planar polynomial autonomous differential systems are introduced and summarized. Meanwhile, the main work of this paper is simply concluded.In Chapter2, we deal with the problem of characterizing center and isochronous centers for complex planar quasi-analytic quardraic system. The technique is based on transforming the quasi-analytic quardraic analytic system into an analytic system. With the help of the computer algebra system-Mathematica, we compute the singular values and period constants of the origin and obtain the necessary center and isochronous center conditions for the transformed system. Finally, we give a proof of the sufficiency by various methods. Our work consists of the existing results related to cubic polynomial system as a special case.In Chapter3, the conditions of center and pseudo-isochronous center conditions at origin for a class of non-analytic septic system are investigated. Firstly, the origin of non-analytic septic system is transferred into the origin of an analytic system by a homeomorphic transformation and a complex transformation. Furthermore, with the help of computer algebra system-Mathematica, we derive the first55singular point quantities at origin of new system and get the center conditions at the origin. Finally, we find necessary conditions for pseudo-isochronous centers by computing its period constants, then the sufficiency of these conditions are proved by some effective methods.In Chapter4, the conditions of center and pseudo-isochronous center conditions at infinity for a class of non-analytic septic system are investigated. Firstly, the infinity is transferred into the origin by a homeomorphic transformation and a complex transfor-mation. Furthermore, with the help of computer algebra system-Mathematica, we derive the first77singular point quantities at infinity and get the center conditions at infinity. Finally, we find necessary conditions for pseudo-isochronous centers by computing its period constants, then the sufficiency of these conditions are proved by some effective methods. In Chapter5, for the three-order nilpotent critical point of classes of quardraic,quintic, septic Lyapunov systems, the center problem and bifurcation of limit cycles are inves-tigated. With the help of computer algebra system-Mathematica, the first11,12,14quasi-Lyapunov constants are deduced respectively. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact of there exist11,12,14small amplitude limit cycles created from the three-order nilpotent critical point is also proved respectively.In Chapter6, we give a method to compute the singular values for a class of Lienard system. By using this new method, the center conditions and bifurcations of limit circles are investigated for a special Lienard system.In Chapter7, we are interested in a cubic Kolmogrov system, the first five singular values are calculated, and the the center conditions are obtained and proved by some technical transformations. Furthermore, five limit circles could be bifurcated from the neighborhood of the origin.