Two Kinds Of Generic About Hamiltonian Systems

Finding the lowest upper bound for the number of the isolated zeros of Abelian integrals, which is called weakened Hilbert 16th problem,is an important problem in bifurcation theory of ordinary differential equation.It is closely related to determining the number of limit cycles of a polynomial perturbated Hamiltonian system or integrable system on the plane.A class of Hamil-tonian systems with special properties,which is called Generic,is easier to be studied,because it has some good nature.This paper consists of three chapters.Chapter 1 is a introduction about the situation of study, definitions of n-Generic and M-Generic and the main conclusions of this paper.In chapter 2,a re-sult is given that quadric Hamiltonian systems with a family of periodic closed orbits must have a center singularity. A class of cubic Hamiltonian systems,H(x,y)=(?) Dy3,where A.B.C.D are real parameters,is studied.It is given that the sufficient and necessary condition of which the corresponding systems are 2-Generic.The condition is different from the results of Il'yashenko and E.Horozov,I.D.Iliev,and two examples are stated to expressed it. In chapter 3,another Generic,non-degenerate semi-weighted homogeneous polynomial,is called M-Generic,because it refers to Petrov modules.In bibliography[16],Gavrilor points that a class of polynomials,such as H(x,y)=y2＋P(x),is non-degenerate semi-weighted homogeneous poly-nomials without strict proof.In the chapter, the ambiguous conclusion is analyzed and discussed first.Then classes of special polynomials are analyzed,and the upper bound of the number of generators of Petrov modules,which are generated by them,is received.