Probabilistic control of chaos: Chaotic maps under control

Recently, we have proposed a new probabilistic method for the control of chaotic systems [1]. In this paper, we apply our method to characteristic cases of chaotic maps (one-and two-dimensional examples). As these chaotic maps are structurally stable, they cannot be controlled using conventional control methods without significant change of the dynamics. Our method consists in the probabilistic coupling of the original system with a controlling system. This coupling can be understood as a feedback control. of probabilistic nature. The chosen periodic orbit of the original system is a global attractor for the probability densities. The generalized spectral decomposition of the associated Frobenius-Perron operator provides a spectral condition of controllability for chaotic dynamical systems.