  We build a smooth real potential $V(t,x)$ on $(t_0,+\infty)\times
\mathbb{R}^2$ decaying to zero as $t\to \infty$ and a smooth solution to the
associated perturbed cubic Nonlinear Harmonic Oscillator whose Sobolev norms
blow up logarithmically as $t\to \infty$. Adapting the method of Faou and
Raphael for the linear case, we modulate the solitons associated to the
Nonlinear Harmonic Oscillator by time-dependent parameters solving a
quasi-Hamiltonian dynamical system whose action grows up logarithmically, thus
yielding logarithmic growth for the Sobolev norm of the solution.
