Partial Control of a System with Fractal Basin Boundaries

In this paper we apply the technique of partial control of a chaotic system to a dynamical system with two attractors with fractal basin boundaries, in presence of environmental noise. This technique allows one to keep the trajectories far from any of the attractors by applying a control that is smaller than the amplitude of environmental noise. We will show that the same geometrical horseshoe-like action that gives rise to the existence of fractal basin boundaries will allow us to detect certain sets for the dynamical system considered, the safe sets, that make this type of control possible.