Parametric partial control of chaotic systems

Discrete dynamical systems where one or several of their parameters vary randomly every iteration are usually referred to as random maps in the literature. However, very few methodologies have been proposed to control these kinds of systems when chaos is present. Here, we propose an extension of the partial control method, that we call parametric partial control, that can be naturally applied to random maps. We show that using this control method it is possible to avoid escapes from a region of the phase space with a transient chaotic behavior. The main advantage of this method is that it allows to control the system even if the corrections applied to the parameter are smaller than the disturbances affecting it. To illustrate how the method works, we have applied it to three paradigmatic models in nonlinear dynamics, the logistic map, the Henon map and the Duffing oscillator.