Further progress in partial control of chaotic systems

The Partial Control technique is a new paradigm in the control of chaotic systems in the presence of noise. This new approach of control allows one to keep trajectories of a dynamical system inside a region from which nearly all the trajectories diverge. Its main advantage is that this goal is achieved even if the corrections applied to the trajectories (control) are smaller than the action of environmental disturbances (noise) on the dynamics. This is a rather counterintuitive result, that is achieved thanks to what we call safe sets. Here, we study the use of the Partial Control technique in one of the most famous chaotic maps, the Hˇznon map, and the deep relationship between the safe sets and the sets of points with different escape times, the escape time sets. We also show how it is possible to find certain extended safe sets that can be used instead of the safe sets in the Partial Control technique. We also discuss briefly the development of a new algorithm to find these new safe sets in any dynamical system with a chaotic behaviour.