  We consider the gravity-capillary water waves equations for a bi-dimensional
fluid with a periodic one-dimensional free surface. We prove a rigorous
reduction of this system to Birkhoff normal form up to cubic degree. Due to the
possible presence of 3-waves resonances for general values of gravity, surface
tension and depth, such normal form may be not trivial and exhibit a chaotic
dynamics (Wilton-ripples). Nevertheless we prove that for all the values of
gravity, surface tension and depth, initial data that are of size $\epsilon$ in
a sufficiently smooth Sobolev space lead to a solution that remains in an
$\epsilon$-ball of the same Sobolev space up to times of order $\epsilon^{-2}$.
We exploit that the $3$-waves resonances are finitely many, and the Hamiltonian
nature of the Birkhoff normal form.
