Types Of System To Be Analytical And Other Center And The Limit Cycle Branch

This thesis probes into the center conditions, the isochronous conditions and bifurcation of limit cycles for quasi-analytic systems of origin. The thesis can be divided into three parts.In chapter one, historical background and research status of the isochronous center and bifurcation of limit cycles of planar polynomial differential system are explained and summarized.Chapter two explores the some aspects concerning the center condition and isochronous center of quasi-fourth system of origin. By means of a series of conversions, the quasi-analytic system is transformed into a complex system. The thesis presents the recursion formulas for computation of singular point quantities and period constants, and deduces the first twelve singular point quantities through the Mathematica system in the computer so as to arrive at the conclusion that the system origin is the necessary and sufficient condition for the center. Then, on the basis of center conditions, and through the computation of period constants, the thesis proposes that the center is the necessary condition for isochronous center. The sufficiency of these conditions is proved through some effective methods.In chapter three, by taking advantage of the method explained in chapter two, center conditions and bifurcation of limit cycles of a class of quasi-fifth system of origin are investigated, and the first fourteen singular point quantities are deduced. In this way, the conditions for the origin to be a center and fourteen-order fine focus are derived respectively. And on this foundation, an example demonstrating that the system can bifurcate five limit cycles at the origin without constructing Poincarécycle domains is demonstrated.