Computing Domains of Attraction for Planar Dynamics

In this note we investigate the problem of computing the domain of attraction of a flow on R-2 for a given attractor. We consider all operator that takes two inputs, the description of the flow and a cover of the attractors, and outputs the domain of attraction for the given attractor. We show that: (i) if we consider only (structurally) stable systems, the operator is (strictly semi-)computable; (ii) if we allow all systems defined by C-1-functions, the operator is not (semi-)computable. We also address the problem of computing limit cycles on these systems.