Adaptive feedback control of periodic orbits in chaotic systems

We describe an adaptive feedback control technique that is able to properly force the evolution of a chaotic system toward a desired periodic motion. First, we present a method enabling chaotic systems to change their dynamics to a stable periodic orbit, based on an adaptive feedback adjustment of an additional parameter of the system, and we discuss its reliability by applying it to several discrete-time systems, mainly focusing on one-dimensional unimodal maps. Then, we propose a strategy for controlling periodic orbits of desired periods in a chaotic dynamics and tracking them toward the set of unstable periodic orbits embedded within the original chaotic attractor. The proposed strategy does not require information on the system to be controlled nor on any reference states for the targets. Assessments on the method's effectiveness and robustness are given by means of the application of the technique for the stabilization of unstable periodic orbits in both discrete- and continuous-time systems. Additionally, we show how this procedure can be exploited to visualize the bifurcation structures of a chaotic dynamical system.