Periodic orbits of a perturbed 3-dimensional isotropic oscillator with axial symmetry

We study the periodic orbits of a generalized Yang–Mills Hamiltonian H depending on a parameter β . Playing with the parameter β we are considering extensions of the Contopoulos and of the Yang–Mills Hamiltonians in a 3-dimensional space. This Hamiltonian consists of a 3-dimensional isotropic harmonic oscillator plus a homogeneous potential of fourth degree having an axial symmetry, which implies that the third component N of the angular momentum is constant. We prove that in each invariant space H = h > 0 the Hamiltonian system has at least four periodic solutions if either β < 0 , or β = 5 + 13 ; and at least 12 periodic solutions if β > 6 and β ≠ 5 + 13 . We also study the linear stability or instability of these periodic solutions.