Limits of the Formal Integrals of Motion
We consider a formal (approximate) integral of motion in Hamiltonians of the form $H=\frac{1}{2}(X^2+Y^2+ω_1^2x^2+ω_2^2y^2)+ε(ηxy^2+αx^3+βx^2y+γy^3)$ generalizing previous cases with $β=γ=0$. First we give the general form of this integral when $ω_1/ω_2$ is irrational and then we consider the case of commensurable frequencies. In particular we study the integrals for the resonances $ω_1/ω_2=4/1, 5/1, 3/2, 4/3, 3/1$ and $2/1$. We also calculate the invariant curves and the orbits in the cases $ω_1/ω_2=2/1$ and $1/1$ (with $β=γ=0$) and we compare the exact-numerical and the theoretical results predicted by the formal integral when $βγ\neq0$. In the special case $ω_1/ω_2=1/1$ we find an integral when $β=γ=0$ and $ηα\neq0$ or $η=α=0$ and $βγ\neq 0$, but this is not possible when $ηαβγ\neq 0$. However, we find that the invariant curves and the orbits can be approximated by a non-resonant integral with $ω_1/ω_2=5\sqrt{2}/7=1.010\dots$.