Let $H:P \to \mathbb{R}$ be a Hamiltonian on a symplectic manifold $(\omega,P)$ and let $X_H: P \to TP$ be the Hamiltonian vectorfield. Let $F:P \to T^*P$ be a dissipative force field such that for $Y = \omega^{\sharp}(F)$ we have that $Y[H] < 0$ everywhere outside a point $x_0 \in P$. This makes $x_0$ a stable point of the dissipative Hamiltonian system $(P,\omega,H,F)$. Now let $f: \mathbb{R} \times P \to T^*P$ be a timeperiodic force. My question: Does there exists a periodic orbit (near $x_0$) for the periodically forced system $(P,\omega,H, F + \epsilon f)$ for sufficiently small $\epsilon$ ? I'm sure the answer is yes, but how big can $\epsilon$ be?
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