A simple global representation for second-order normal forms of Hamiltonian systems relative to periodic flows

We study the determination of the second-order normal form for perturbed Hamiltonians $H_{ϵ}=H_0+ϵH_1+\frac{{ϵ}^2}{2}{H}_2$, relative to the periodic flow of the unperturbed Hamiltonian $H_0$. The formalism presented here is global, and can be easily implemented in any CAS. We illustrate it by means of two examples: the H'enon-Heiles and the elastic pendulum Hamiltonians.