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Abstract: We investigate the computational power of several models of dynamical systems under infinitesimal perturbations of their dynamics. We consider in our study models for discrete and continuous time dynamical systems: Turing machines, Piecewise affine maps, Linear hybrid automata and Piecewise constant derivative systems (a simple model of hybrid systems). We associate with each of these models a notion of perturbed dynamics by a small varepsilon (w.r.t. to a suitable metrics), and define the perturbed reachability relation as the intersection of all reachability relations obtained by varepsilon-perturbations, for all possible values of varepsilon. We show that for the four kinds of models we consider, the perturbed reachability relation is co-recursively enumerable, and that any co-r.e. relation can be defined as the perturbed reachability relation of such models. A corollary of this result is that systems that are robust, i.e., their reachability relation is stable under infinitesimal perturbation, are decidable.