Control of Transient Chaos Using Safe Sets in Simple Dynamical Systems

Transient chaos is nearly as ubiquitous as chaos itself, and it is a manifestation of the existence of a nonattractive chaotic set: a chaotic saddle. In some situations it might be desirable to keep the trajectories of a dynamical system with transient chaos far from the attractor and close to this set but its nonattractive nature, the complex dynamics associated with it and eventually the presence of noise may difficult this task. Assume, as an extra difficulty, that our action on the system is bounded and smaller than the action of noise. In such a situation this might seem impossible. However, we will show that in a variety of one dimensional maps this is possible indeed. The control strategy is based on the existence of a set, the safe set, with interesting properties that are due to the same conditions that imply the existence of a chaotic saddle in the system. An example of application of our control technique with the logistic map and some numerical simulations confirming our results are also presented in this work.