The Estimate Of The Exact Number Of Zeros Of Some Abelian Integrals

Using the criteria function methods,this paper gives exact estimates of the number of zeros of some Abelian integrals.The full text is divided into six chapters.The first chapter mainly introduces the background of the number of zeros of Abelian integrals,such as the weakened Hilbert 16th problem,and some existing research statuses.In the second chapter,we introduce some basic concepts and some existing criteria function methods for estimating the number of zeros of Abelian integral.And derive under the same system the advantages and disadvantages of two different criteria function methods.Finally,by using the selected method,we get a class of Lienard system with at most two limit cycles under the corresponding disturbance,But the other method can't solve the problem completely.The third chapter studies a sub-problem of the Bagdanov-Takens branch of nilpotent codimension-3 of singular point.This problem can be transformed into the number of zeros of Abelian integral,and the integrand contain logarithmic function.In this paper.the idea of Picard-Fuchs equation can be used to convert the logarithmic function,so that the corresponding problems can be solved by computer algebra system,Furthermore,we can prove that the corresponding Abelian integral can produce at most two zeros by the method of criteria function.In the fourth chapter,we mainly consider the cyclicity of the periodic annulus of a class of quadratic reversible system,and the first integral of this kind of system has contain the logarithmic function.This problem is also transformed into the number of zeros of Abelian integral,and the integrand contain logarithmic function.In this paper,the study of the corresponding problem can be transformed into a semi-algebraic system by appropriate transformation,and its cyclicity is proved to be 2.Finally,by exploring the geometric properties of the corresponding Abelian integral ratio,all possible limit cycle distributions of this type of system are obtained.In chapter 5,we study the Poincare bifurcation problem of a class of non-smooth oscil-lators.Verification found that some existing criteria function methods cannot completely solve this problem,Furthermore,by improving the existing criteria function method and combining with the behavior of the corresponding Abelian integral at the end point,it can be concluded that this kind of system can generate at most two limit cycles,and the upper bound is reachable.In chapter 6,we mainly discuss the number of isolated zeros of a class of hyperelliptic integrals.Based on the existing criteria function methods can not solve this problem.In this chapter,we give a new criteria function method,and simplify the corresponding discriminant function method under some special conditions,and get the new criteria function method is more effective than the result in Chapter 5.By using this method,we get that when the integral curve of this kind of hyperelliptic integrals is located in the closed curve family of the real form,the number of isolated zeros is at most 2,and this bound is reachable.