Predictive Poincaré control: A control theory for chaotic systems

One of the most interesting features of chaotic systems is the large number of unstable orbits embedded in a chaotic attractor. In this work, we propose a global chaos-control technique called predictive Poincar\'e control (PPC) that permits stabilization of a predefined solution, using only small control pulses. We prove this result for a large class of n-dimensional chaotic systems. The predefined solution can be a periodic or nonperiodic oscillation, expressed by a periodic or nonperiodic symbolic sequence [S. Hayes, C. Grebogi, and E. Ott, Phys. Rev. Lett. 70, 3031 (1993)]. We apply the general PPC scheme to the well known Lorenz model and study its robustness with respect to parasitic effects.