Reversible Critical Period When Other Central Branch And A Class Of Polynomial Systems Limit Ring

One of the main problems in the qualitative theory of the planar differentialsystems is to study the number of critical periods and to study the number anddistribution of their limit cycles. This paper is devoted to studying bifurcation ofcritical periods from the reversible rigidly isochronous centers and finding thenumber of limit cycles of a class polynomial differential systems.In chapter1, we introduce some preliminary knowledge including dynamicsystem and theory of bifurcation, and then, the author briefly introduce the researchwork of this paper.In chapter2, we discuss some basic theory that relate to this paper.In chapter3, we consider the planar system x_=(?y+xFm?1(x, y))(1+εμ(x, y)),y_=(x+yFm?1(x, y))(1+εμ(x, y)), where0<ε?1, and μ(x, y)=Σs+t≤n bstxsyt, n≥2. We will discuss bifurcation of critical periods of this system. First, we will giveexpressions of period bifurcation function in the form of integrals, and apply ourmethod to study the number of critical periods of the linear isochronous vector fieldand the number of critical periods of the quadratic and cubic isochronous vector fieldrespectively.In chapter4, we study the planar system of the form x_=?yC(x, y)+εP(x, y), y_=xC(x, y)+εQ(x, y), where C(x, y)=(1? y2)m, C(0,0)?=0, and ε is a small realparameter.we will discuss the problem of number of limit cycles of the system bysimplifying the abelian integral.