In this paper, we give the different topological types of phase portrait for Liénard system \(\dot {x}=y, \dot {y}=-g(x)\) in the case that \(\deg g(x)=7\) and the system have six and seven singular points, respectively. For its perturbed system, the expansion of the Melnikov function near any of the above closed orbits, except that the closed orbit is a compound loop passing through a nilpotent cusp and two hyperbolic saddles or passing through three hyperbolic saddles, has been studied. In this paper, as one of main results, for a near-Hamiltonian system, we give the expansion of the Melnikov function near a compound loop with a nilpotent cusp and two hyperbolic saddles. Based on this, we present the conditions to obtain limit cycles.

在本文中，我们给出了Liénard系统\（\ dot {x} = y，\ dot {y} =-g（x）\）时相拓扑的不同拓扑类型，其中\（\ deg g（x ）= 7 \），系统分别具有六个和七个奇异点。对于其扰动系统，已经研究了梅尔尼科夫函数在上述任何闭合轨道附近的扩展，除了该闭合轨道是穿过零能尖点和两个双曲线鞍座或穿过三个双曲线鞍座的复合环。在本文中，作为主要结果之一，对于近似哈密顿系统，我们给出了具有零尖点和两个双曲鞍的复合环附近的梅尔尼科夫函数的展开。基于此，我们提出了获得极限循环的条件。