Bifurcation And Center Problems For Two Kind Of Differential Autonomous Systems

This thesis explores the central conditions, bifurcation of limit cycles of origin is the nilpotent singular point, also devoted to quasi-analytic system bifurcation of limit cycles and integrality conditions.First of all, the thesis explains and summarizes the historical back ground and re-search status of the central conditions, bifurcation of limit cycles for the nilpotent system and quasi-analytic system. 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 origin is the nilpotent singular point of the forth systems, with a computation formula, for the singular point quantities of the center provided. At the same time, the first10singular point quantities are deduced by means of Matheniatica in the computer. On this foundation, the thesis comes to the conclusion that this class of forth systems can result in limit Cycles of10small amplitudes at the origin.Finally, by taking advantage of the method explained in Chapter3, central conditions and bifurcation of limit cycles of quasi quintic systems are investigated.In aid of the Mathematica system in the computer, the first24singular point quantities are deduced. And the problems of the conditions of this system are investigated. Meanwhile, the conditions for the origin to be a center and18-order fine focus are derived respectively, and this system can bifurcate6limit cycles.