Integrals and chaos in generalized H\'{e}non-Heiles Hamiltonians

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.