  In this paper we combine two existing approaches for approximating
attractors. One of them approximates the attractors arbitrarily well by
sublevel sets related to solutions of infinite dimensional linear programming
problems. A downside there is that these sets are not necessarily positively
invariant. On the contrary, the second method provides supersets of the
attractor which are positively invariant. Their method on the other hand has
the disadvantage that the underlying optimization problem is not
computationally tractable without the use of heuristics - and incorporating
them comes at the price of losing guaranteed convergence. In this paper we
marry both approaches by combining their techniques and we get converging outer
approximations of the attractor consisting of positively invariant sets based
on convex optimization via sum-of-squares techniques. The method is easy to use
and illustrated by numerical examples.
