本项目拟通过分析哈密顿函数的代数结构、特别哈密顿函数定义的对合映射参数化分析，研究若干低次哈密顿系统n次光滑和非光滑的扰动问题。一是研究低次(超)椭圆哈密顿系统的光滑和非光滑n次Liénard型扰动问题、探索一般n次扰动问题的周期环域环性数；二是研究四次和五次超椭圆哈密顿系统的非充分小的光滑和非光滑n次Liénard型扰动的Hopf环性数；三是研究具有不变直线的三次哈密顿系统的拓扑分类，及其中的连接不变直线上奇点的同异宿环附近的Abel积分渐近展开式和三次扰动环性数下界。

In this project, we will study the annulus cyclicity and Hopf cyclicity in the smooth and non-smooth perturbation of some Hamiltonian systems by investigating the algebraic structure of the Hamiltonians, especially the parameterization of the involutions. Firstly, we will study the annulus cyclicity in some elliptic and hyperelliptic Hamiltonian systems with small smooth and non-smooth perturbation of degree n, including Liénard type and general polynomial perturbation; Secondly, we study the Hopf cyclicity in some quartic and quintic hyperelliptic Hamiltonian systems damped by smooth and non-smooth special polynomials of degree n, which are belonging to generalized Liénard systems, and we inspect the Hopf cyclicity in two elliptic Hamiltonian systems damped by general polynomials; Thirdly, we will study the topological classification of some cubic Hamiltonian systems with an invariant strait line, and the asymptotic expansions of Abelian integrals near the homoclinic loop and heteroclinic loop connecting a singularity on the invariant strait line. In particular, we will give a lower bound of annulus cyclicity by the asmptotic expansions.