We consider time-periodic Hamiltonians of the form \(H(t, {\textbf {Q}}, {\textbf {P}}, \epsilon )\) where \(\epsilon \) is a small parameter: The unperturbed function \(H_0({\textbf {Q}}, {\textbf {P}})=H(t,{\textbf {Q}}, {\textbf {P}}, 0)\) is autonomous, integrable and has periodic solutions. It is assumed that these Hamiltonian functions can be written in convenient symplectic coordinates in the form \(\begin{aligned} H(t,\theta , \phi ,{\textbf {q}}, I, J, {\textbf {p}},\epsilon )=H_0(I,J)+\epsilon H_1(t,\theta , \phi ,{\textbf {q}}, I, J,{\textbf {p}})+\mathcal {O}(\epsilon ^2), \end{aligned}\) where \(\theta ,\phi \in \mathbb {T}\) , \(I,J\in \mathbb {R}\) , \({\textbf {q}}, {\textbf {p}}\in \mathbb {R}^n\) . The aim of this paper is to show the existence of periodic solutions of the previous family of time-dependent \(2\pi \) -periodically perturbed Hamiltonian systems under different approaches.
