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We consider a general class of finite dimensional deterministic dynamicalsystems with finitely many local attractors $K^i$ each of which supports aunique ergodic probability measure $P^i$, which includes in particular theclass of Morse-Smale systems in any finite dimension. The dynamical system isperturbed by a multiplicative non-Gaussian heavy-tailed L\'evy type noise ofsmall amplitude $\varepsilon>0$. Specifically we consider perturbations leadingto a It\^o, Stratonovich and canonical (Marcus) stochastic differentialequation. The respective asymptotic first exit time and location problem fromeach of the domains of attractions $D^i$ in case of inward pointing vectorfields 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 withcharacteristic boundaries and show that the perturbed system exhibits ametastable behavior in the sense that there exits a unique$\varepsilon$-dependent time scale on which the random system converges to acontinuous time Markov chain switching between the invariant measures $P^i$. Asexamples we consider $\alpha$-stable perturbations of the Duffing equation anda chemical system exhibiting a birhythmic behavior.

Metastability of Morse-Smale dynamical systems perturbed by heavy-tailed L\'evy type noise