We study arbitrary cubic perturbations of the symmetric 8-loop Hamiltonian, which are linear with respect to the small parameter. It is shown that when the first 4 coefficients in the expansion of the displacement functions corresponding to both period annuli inside the loop vanish, the system becomes integrable with the following three strata in the center manifold: Hamiltonian, reversible in $y$ and reversible in $x$. In the latter case, the first integral is of Darboux type and we calculate it explicitly. Next we prove that the cyclicity of each of period annuli inside the loop is five and the total cyclicity of both is at most nine. For this, we use Abelian integrals method together with careful study the geometry of the separatrix solutions of related Riccati equations in connection to the second-order Melnikov functions.

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关于对称八环哈密顿量的三次摄动

我们研究对称 8 环哈密顿量的任意三次扰动，它们与小参数呈线性关系。结果表明，当循环内对应于两个周期环的位移函数展开的前 4 个系数消失时，系统变得与中心流形中的以下三个层可积：Hamiltonian，在 $y$ 中可逆和在 $y$ 中可逆$x$。在后一种情况下，第一个积分是 Darboux 类型的，我们明确计算它。接下来我们证明循环内每个周期环的周期为 5，两者的总周期最多为 9。为此，我们使用阿贝尔积分方法并仔细研究了与二阶 Melnikov 函数相关的相关 Riccati 方程的分离矩阵解的几何形状。