标题： 积分和广义Hénon-Heiles哈密顿系统中的混沌 显示英文标题

标题： Integrals and chaos in generalized Hénon-Heiles Hamiltonians

摘要： 我们研究哈密顿量$ H = \frac{1}{2}\left( \dot{x}^2 + \dot{y}^2 + x^2 + y^2 \right) + \epsilon\,\left( xy^2 + \alpha x^3\right)$的近似（形式）运动积分，该哈密顿量是通常的 Hénon-Heiles 哈密顿量的扩展，具有$\alpha = -1/3$。 我们比较理论截面图（在$y=0$）与通过数值积分许多轨道计算出的精确截面图。 对于小的$\epsilon$，理论截面图和精确截面图的不变曲线彼此接近，但对于大的$\epsilon$存在差异。 最重要的是在精确情况下出现混沌，当$\epsilon$接近$\alpha<0$的逃逸扰动时，这种混沌变得占主导地位。 我们特别研究了$\alpha = 1/3$的情况，它表示一个可积系统，以及$\alpha = 0$。 最后我们通过共振重叠机制在$\alpha=-1/3$的情况下（原始的Henon-Heiles系统）进行了检验，通过显示不稳定周期轨道的渐近曲线的同宿和异宿交叉来说明混沌的产生。 显示英文摘要

摘要： We study the approximate (formal) integrals of motion in the Hamiltonian $ H = \frac{1}{2}\left( \dot{x}^2 + \dot{y}^2 + x^2 + y^2 \right) + \epsilon\,\left( xy^2 + \alpha x^3\right)$ which is an extension of the usual H\'{e}non-Heiles Hamiltonian that has $\alpha = -1/3$. We compare the theoretical surfaces of section (at $y=0$) with the exact surfaces of section calculated by integrating numerically many orbits. For small $\epsilon$, the invariant curves of the theoretical and the exact surfaces of section are close to each other, but for large $\epsilon$ there are differences. The most important is the appearance of chaos in the exact case, which becomes dominant as $\epsilon$ approaches the escape perturbation for $\alpha<0$. We study in particular the cases $\alpha = 1/3$, which represents an integrable system, and $\alpha = 0$. Finally we examine the generation of chaos through the resonance overlap mechanism in the case $\alpha=-1/3$ (the original H\'{e}non-Heiles system) by showing both the homoclinic and the heteroclinic intersection of the asymptotic curves of the unstable periodic orbits.