Hamiltonian oscillators in \(1\)-\(1\)-\(1\) resonance: Normalization and integrability
Here, a family of three degree of freedom Hamiltonian systems defined by a Taylor expansion around an elliptic equilibrium is considered. 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. A second-order normal form using the invariants of the reduced phase space is obtained. The normal form is expressed in the invariants and in only six terms. By means of the nodal-Lissajous variables, relations are obtained among the parameters defining the perturbation, which lead to integrable subfamilies. Finally some applications are given.
scientific article; zbMATH DE number 1439883