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We consider a finite dimensional deterministic dynamical system with a globalattractor A with a unique ergodic measure P concentrated on it, which isuniformly parametrized by the mean of the trajectories in a bounded set Dcontaining A. We perturbe this dynamical system by a multiplicative heavytailed L\'evy noise of small intensity \epsilon>0 and solve the asymptoticfirst exit time and location problem from a bounded domain D around theattractor A in the limit of {\epsilon} to 0. In contrast to the case ofGaussian perturbations, the exit time has the asymptotically algebraic exitrate as a function of \epsilon, just as in the case when A is a stable fixedpoint. In the small noise limit, we determine the joint law of the first timeand the exit location on the complement of D. As an example, we study the firstexit problem from a neighbourhood of a stable limit cycle for the Van der Poloscillator perturbed by multiplicative \alpha-stable L\'evy noise.

The exit problem from the neighborhood of a global attractor for heavy-tailed L\'evy diffusions