  We study the connection between the appearance of a `metastable' behavior of
weakly chaotic orbits, characterized by a constant rate of increase of the
Tsallis q-entropy (Tsallis 1988), and the solutions of the variational
equations of motion for the same orbits. We demonstrate that the variational
equations yield transient solutions, lasting for long time intervals, during
which the length of deviation vectors of nearby orbits grows in time almost as
a power-law. The associated power exponent can be simply related to the
entropic exponent for which the q-entropy exhibits a constant rate of increase.
This analysis leads to the definition of a new sensitive indicator
distinguishing regular from weakly chaotic orbits, that we call `Average Power
Law Exponent' (APLE). We compare the APLE with other established indicators of
the literature. In particular, we give examples of application of the APLE in
a) a thin separatrix layer of the standard map, b) the stickiness region around
an island of stability in the same map, and c) the web of resonances of a 4D
symplectic map. In all these cases we identify weakly chaotic orbits exhibiting
the `metastable' behavior associated with the Tsallis q-entropy.
