  The superdiffusion behavior, i.e. $<x^2(t)> \sim t^{2 \nu}$, with $\nu >
1/2$, in general is not completely characherized by a unique exponent. We study
some systems exhibiting strong anomalous diffusion, i.e. $<|x(t)|^q> \sim t^{q
\nu(q)}$ where $\nu(2)>1/2$ and $q \nu(q)$ is not a linear function of $q$.
This feature is different from the weak superdiffusion regime, i.e.
$\nu(q)=const > 1/2$, as in random shear flows. The strong anomalous diffusion
can be generated by nontrivial chaotic dynamics, e.g. Lagrangian motion in $2d$
time-dependent incompressible velocity fields, $2d$ symplectic maps and $1d$
intermittent maps. Typically the function $q \nu(q)$ is piecewise linear. This
corresponds to two mechanisms: a weak anomalous diffusion for the typical
events and a ballistic transport for the rare excursions. In order to have
strong anomalous diffusion one needs a violation of the hypothesis of the
central limit theorem, this happens only in a very narrow region of the control
parameters space.
