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

Abstract

We study the periodic orbits of a generalized Yang–Mills Hamiltonian $$\mathcal {H}$$ depending on a parameter $$\beta $$ . Playing with the parameter $$\beta $$ 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 $$\mathcal {H}=h>0$$ the Hamiltonian system has at least four periodic solutions if either $$\beta <0$$ , or $$\beta = 5+\sqrt{13}$$ ; and at least 12 periodic solutions if $$\beta >6$$ and $$\beta \ne 5+\sqrt{13}$$ . We also study the linear stability or instability of these periodic solutions.