We analytically study the Hamiltonian system in $ \mathbb{R}^6$ with Hamiltonian
$H = 1/2 (p_x^2+p_y^2+p_z^2)+\frac{1}{2} (ω_1^2 x^2+ω_2^2 y^2+ ω_3^2 z^2)+ \varepsilon(a z^3 + z (b x^2 +c y^2)),$
being $ a,b,c∈\mathbb{R}$ with $ c\ne 0$, $ \varepsilon$ a small parameter, and $ ω_1$, $ ω_2$ and $ ω_3$the unperturbed frequencies of the oscillations along the $ x$, $ y$ and $ z$ axis, respectively. For $ |\varepsilon|>0$ small, using averaging theory of first and second order we find periodic orbits in every positive energy level of $ H$ whose frequencies are $ ω_1 = ω_2 = ω_3/2$ and $ ω_1 = ω_2 = ω_3$, respectively (the number of such periodic orbits depends on the values of the parameters $ a,b,c$). We also provide the shape of the periodic orbits and their linear stability.
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Figure 2.
Examples of the intersection of the regions
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