  We consider the nonlinear cubic Wave, the Hartree and the nonlinear cubic
Beam equations on $T^2$ and we prove the existence of different types of
solutions which exchange energy between Fourier modes in certain time scales.
This exchange can be considered \emph{chaotic-like} since either the choice of
activated modes or the time spent in each transfer can be chosen randomly. The
key point of the construction of those orbits is the existence of heteroclinic
connections between invariant objects and the construction of symbolic dynamics
(a Smale horseshoe) for the Birkhoff Normal Form truncation of those equations.
