On sample-based computations of invariant sets

In this paper, the classical problem of uncovering the maximal invariant set of a (discrete-time) dynamical system is illuminated from a novel perspective, which in particular leads to a novel sample-based computational procedure to compute the invariant set. The mathematical description of these new insights can be formulated in strikingly basic set-theoretic terms, and more importantly, be efficiently realized computationally in terms of different sample-based implementations. We illustrate the simplicity and efficiency of the computational method on three examples with a maximal invariant set that is unstable in both time directions, the classical Hénon map, a three-dimensional analogue of the Hénon map, and a Van der Pol oscillator.