Periodic orbits of perturbed non–axially symmetric potentials in 1:1:1 and 1:1:2 resonances

We analytically study the Hamiltonian system in R⁶ with Hamiltonian H =₂¹ (p²_x + p²_y + p²_z) +¹₂ (ω₁²x² + ω₂²y² + ω₃²z²) + ε(az³ + z(bx² + cy²)), being a, b, c ∈ R with c 6= 0, ε a small parameter, and ω1, ω₂ and ω₃ the unperturbed frequencies of the oscillations along the x, y and z axis, respectively. For |ε| > 0 small, using averaging theory of first and second order we find periodic orbits in every positive energy level of H whose frequencies are ω1 = ω2 = ω3/2 and ω1 = ω2 = ω3, 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.