本课题利用微分方程定性理论中的经典方法以及最新方法，结合数学机械化中的符号计算技术，研究若干可积系统的扰动分支问题及其应用。一方面研究若干（m，m-1）型Lienard系统所对应的Abel积分零点个数，寻求上确界；研究若干(m,n)型Lienard系统极限环分支个数下界；研究由上述系统经过"三倍复制"得到平面系统的极限环分支，寻求新的Hilbert数的下界以及极限环新分布，另一方面，推广发展研究上述系统的经典方法，寻求新结果，拓广这些方法的应用领域，研究新问题。把研究Abel积分的经典方法和新方法迁移到研究一类受扰动非线性波方程的动力学行为，如孤立周期波解的个数和分布，孤立子的存在与保持性，以及迁移到几类金融系统的分支研究。

We will applly the classical and new methods in the qualitative theory of ordinary differential equations to study some perturbed integrable planar systems, with the help of symbolic computations in mathematics mechanization. First,we study the sharp bound of number of zeros of Abelian integral for some Lienard systems of type (m,m-1); We will also investigate the lower bound of number of limit cycles for some Lienard systems of type (m,n); When duplicating the Lienard systems of type (m,m-1), it become general planar systems, we will investigate their limit cycles for some new Hilbert numbers and distributions. Second, we want to develop the classical methods for new results, and broaden its applications to new problems in other field. For detail, we applly the classical and new methods for abelian integral to study the danamics of some perturbed nonlinear wave equations, such as the number of isolated periodic wave solutions and their distribution, the existence and persistence of solitions, to study some finance systems.