The Center Of The Class Of Differential Systems, The Bifurcation Of Limit Cycles And When The Center

This thesis explores the central conditions, the bifurcation of limit cycles as well as the isochronous conditions in planar polynomial differential systems.First of all, the thesis explains and summarizes the historical background and research status of the central conditions, bifurcation of limit cycles and isochronous conditions of the planar polynomial differential systems. And then a brief introduction to the whole research of the thesis is given.Secondly, the central conditions and bifurcation of limit cycles for a class of quintic systems are probed into, with a computation formula for the singular point quantities of the center provided. At the same time, the first 14 singular point quantities are deduced by means of Mathematica in the computer. On the foundation of the 14 singular point quantities without the construction of Poincare annual domain, the thesis comes to the conclusion that this class of quintic systems can result in limit cycles of 8 small amplitudes at the origin.Finally, by taking advantage of the method explained in Chapter 2, central conditions, isochronous condition and bifurcation of limit cycles of quasi quintic systems are investigated. In aid of the Mathematica system in the computer, the first 11 singular point quantities are deduced. And the problems of the isochronous conditions of this system are investigated. By using a new method, the recursion formulas for calculating periodic constants are given. Then the author proposes the sufficient and necessary conditions for center to be an isochronous center. Meanwhile, the conditions for the origin to be a center and 10-order fine focus are derived respectively, and this system can bifurcate 7 limit cycles.