Hamiltonian systems in 1:2:ω-resonance (ω = 5 or 6)

Abstract

We consider conservative Hamiltonian systems in three degrees of freedom with a combined first and second order resonance, the 1:2:5-resonance and the 1:2:6-resonance. The systems are expanded in a Taylor-series around a stable equilibrium, which we truncate to obtain a finite series. Such systems are integrable in the first approximation, but one integral vanishes when we introduce higher order terms; we then have full interaction between the three degrees of freedom. In these cases we locate the periodic orbits and their stability type and—when possible—give an estimate on the asymptotic validity of the approximations of these orbits (Sections 2 and 3). In the last section we consider the effect of discrete symmetry conditions on our systems.