Dynamics of partial control.

Safe sets are a basic ingredient in the strategy of partial control of chaotic systems. Recently we have found an algorithm, the sculpting algorithm, which allows us to construct them, when they exist. Here we define another type of set, an asymptotic safe set, to which trajectories are attracted asymptotically when the partial control strategy is applied. We apply all these ideas to a specific example of a Duffing oscillator showing the geometry of these sets in phase space. The software for creating all the figures appearing in this paper is available as supplementary material.