Limit Cycles Of Two Classes Of Perturbed Quadratic Reversible Systems Of Genus One

The Hilbert’s 16th problem has always been a hot research topic at home and abroad,and now some corresponding progress has been made.In addition,the key to solve the linear state of the orbit of the plane dynamic system is to study the maximum number of limit cycles and its position relationship,and the importance of the maximum number of limit cycles is obvious in the practical application problems.In this thesis,the qualitative theory of differential equation is taken as the research basis,and the Picard-Fuchs equation method and Riccati equation method are used to study the number of limit cycles that can be generated by two kinds of quadratic reversible systems whose deficiency is one under the perturbation of polynomials of degree n.Firstly,the two kinds of quadratic reversible systems are transformed into their standard forms.Secondly,the integral function is introduced for further study,so that a series of calculations can be carried out to obtain the relation of their Abelian integrals.Then,the Picard-Fuchs equations and Riccati equations are obtained by further using these two methods.That is,the differential equations satisfied between Abelian integrals and their derivatives.Finally,by using some relevant theorems,the upper bound of the number of zeros of Abelian integrals corresponding to them can be obtained,and then the upper bound of the number of limit cycles produced by these two kinds of systems can be obtained.Through the research results of this paper,it can be found that when the perturbation term is in the form of polynomial of higher order,the number of zeros of the relevant Abelian integrals can be obtained by using Picard-Fuchs equation method and Riccati equation method.That is to say,in this paper,we study when n≥4,two kinds of two reversible system of Abelian integrals upper limit of the number of zeros are respectively 2[(n-1)/3]+[(n-2)/3]+4[n/4]+2 and 6[(n-1)/4]+[(n-2)/4]+8[n/4]+3,so when n≥4,the upper bounds of the number of limit cycles for two kinds of quadratic invertible systems are respectively 2[(n-1)/3]+[(n-2)/3]+4[n/4]+2 and 6[(n-1)/4]+[(n-2)/4]+8[n/4]+3.