The Research On Singular Point Quantities And Integrability Problems For Several Types Of Planar Polynomial Differential Systems

This thesis is composed of four chapters, center-focus problem, singular point quantitiesand integrability conditions for some planar polynomial differential systems are mainlystudied.In the first chapter, the historical background and the present progress of the relatedproblems about limit cycles and center-focus of the qualitative theory of planar polynomialdifferential systems are introduced and summarized. At the same time, I make a briefintroduction to this article.In the second chapter, the author introduces some basic concepts, theorems and priorknowledge of the qualitative theory of differential equation.In the third chapter, we meticulously study a class of complex planar cubic system. Wehave given the recursive formula to calculate the origin singular point quantities, and withthis formula, the first6singular point quantities of the system for the origin are computed bycomputer algebra system Mathematica. Then we make use of invariant algebraic curvestheory to obtain the integrability conditions for the origin of the system.In the fourth chapter, a special class of real planar four system is cleverly constructed, itis first transformed into the corresponding complex adjoint system, and then throughcalculating the algebraic recursion formulas of the origin singular point quantities of thesystem, we get the first10singular point quantity expressions for this system at the origin,Further, we discuss the integrability problem of the system. Thus the calculation of the focalquantities and the center-focus decision problems for the real system are equivalencecompleted.Finally, I give out Mathematica machine deducing process and the results for singularpoints quantities of my thesis studies two types of differential systems in the appendix.