  We consider a finite dimensional deterministic dynamical system with a global
attractor A with a unique ergodic measure P concentrated on it, which is
uniformly parametrized by the mean of the trajectories in a bounded set D
containing A. We perturbe this dynamical system by a multiplicative heavy
tailed L\'evy noise of small intensity \epsilon>0 and solve the asymptotic
first exit time and location problem from a bounded domain D around the
attractor A in the limit of {\epsilon} to 0. In contrast to the case of
Gaussian perturbations, the exit time has the asymptotically algebraic exit
rate as a function of \epsilon, just as in the case when A is a stable fixed
point. In the small noise limit, we determine the joint law of the first time
and the exit location on the complement of D. As an example, we study the first
exit problem from a neighbourhood of a stable limit cycle for the Van der Pol
oscillator perturbed by multiplicative \alpha-stable L\'evy noise.
