  This paper develops a method for obtaining guaranteed outer approximations
for global attractors of continuous and discrete time nonlinear dynamical
systems. The method is based on a hierarchy of semidefinite programming
problems of increasing size with guaranteed convergence to the global
attractor. The approach taken follows an established line of reasoning, where
we first characterize the global attractor via an infinite dimensional linear
programming problem (LP) in the space of Borel measures. The dual to this LP is
in the space of continuous functions and its feasible solutions provide
guaranteed outer approximations to the global attractor. For systems with
polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares
tightenings of the dual LP provides a sequence of outer approximations to the
global attractor with guaranteed convergence in the sense of volume discrepancy
tending to zero. The method is very simple to use and based purely on convex
optimization. Numerical examples with the code available online demonstrate the
method.
