Periodic motion in perturbed elliptic oscillators revisited

Abstract

We analytically study the Hamiltonian system in $\mathbb{R}^{4}$ with Hamiltonian

$$\begin{aligned} H= \frac{1}{2} \bigl(p_{x}^{2}+p_{y}^{2} \bigr)+\frac{1}{2} \bigl(\omega_{1}^{2} x ^{2}+\omega_{2}^{2} y^{2} \bigr)- \varepsilon V(x,y) \end{aligned}$$

being $V(x,y)=-(x^{2}y+ax^{3})$ with $a\in\mathbb{R}$, where $\varepsilon$ is a small parameter and $\omega_{1}$ and $\omega_{2}$ are the unperturbed frequencies of the oscillations along the $x$ and $y$ axis, respectively. Using averaging theory of first and second order we analytically find seven families of periodic solutions in every positive energy level of $H$ when the frequencies are not equal. Four of these seven families are defined for all $a\in\mathbb{R}$ whereas the other three are defined for all $a\ne0$. Moreover, we provide the shape of all these families of periodic solutions. These Hamiltonians may represent the central parts of deformed galaxies and thus have been extensively used and studied mainly numerically in order to describe local motion in galaxies near an equilibrium point.