Periodic orbits in analytically perturbed Poisson systems

• Finite-dimensional Poisson systems are present in most fields of physics.
• We show that some Poisson systems are orbital equivalent to a linear Darboux canonical form.
• After perturbations we analyze the bifurcation phenomena of periodic orbits.
• We apply the technique to several interesting oscillators.

Analytical perturbations of a family of finite-dimensional Poisson systems are considered. It is shown that the family is analytically orbitally conjugate in U⊂RnU⊂Rn to a planar harmonic oscillator defined on the symplectic leaves. As a consequence, the perturbed vector field can be transformed in the domain UU to the Lagrange standard form. On the latter, use can be made of averaging theory up to second order to study the existence, number and bifurcation phenomena of periodic orbits. Examples are given ranging from harmonic oscillators with a potential and Duffing oscillators, to a kind of zero-Hopf singularity analytic normal form.