Hamiltonian oscillators in 1-1-1 resonance:: Normalization and integrability

We consider a family of three-degree-of-freedom (3-DOF) Hamiltonian systems defined by a Taylor expansion around an elliptic equilibrium. More precisely, the system is a perturbed harmonic oscillator in 1-1-1 resonance. The perturbation is an arbitrary polynomial with cubic and quartic terms in Cartesian coordinates. We obtain the second-order normal form using the invariants of the reduced phase space. This normal form is defined by six quantities that correspond to the interaction terms associated to this resonance. Then, by means of the nodal-lissajous variables, we obtain relations among the parameters defining the perturbation, which lead to integrable subfamilies. Finally some applications are given.