  We consider a general class of finite dimensional deterministic dynamical
systems with finitely many local attractors $K^i$ each of which supports a
unique ergodic probability measure $P^i$, which includes in particular the
class of Morse-Smale systems in any finite dimension. The dynamical system is
perturbed by a multiplicative non-Gaussian heavy-tailed L\'evy type noise of
small amplitude $\varepsilon>0$. Specifically we consider perturbations leading
to a It\^o, Stratonovich and canonical (Marcus) stochastic differential
equation. The respective asymptotic first exit time and location problem from
each of the domains of attractions $D^i$ in case of inward pointing vector
fields in the limit of $\varepsilon \to 0$ was solved by the authors in [J.
Stoch. An. Appl. 32(1), 163-190, 2014]. We extend these results to domains with
characteristic boundaries and show that the perturbed system exhibits a
metastable behavior in the sense that there exits a unique
$\varepsilon$-dependent time scale on which the random system converges to a
continuous time Markov chain switching between the invariant measures $P^i$. As
examples we consider $\alpha$-stable perturbations of the Duffing equation and
a chemical system exhibiting a birhythmic behavior.
