Chapter 24

Chaotic attractors contain unstable periodic orbits of any desired period (this is shown in chapter 22 for Axiom A attractors). Furthermore, for an ergodic attractor we know that any trajectory will eventually come arbitrarily close to any of these orbits. This offers the opportunity for controlling chaos: when the chaotic orbit approaches the unstable periodic orbit of interest it can be attracted to and maintained on the orbit by applying small perturbations to the system. There are two aspects to the problem. The first, depending on the properties of the whole attractor, is the idea that waiting long enough guarantees that the orbit will come arbitrarily close to any chosen point on the attractor. Alternatively knowledge of the chaotic attractor can be used to speed this process by directing the orbit to the desired region—this idea is studied further in the next chapter. The second part is to use delicate perturbations of the system to keep the orbit on or very close to the unstable orbit. This part can be analyzed in terms of small perturbations from the orbit, i.e. by a linear analysis. There is an enormous literature in “Control Theory” in engineering that can be taken over for this second part. An intriguing aspect of control in the context of chaotic systems is that different periods can be selected simply by changing the nature of the delicate perturbations of the system. The idea of “controlling chaos” was suggested in a famous paper by Ott, Grebogi and Yorke [1], and we will first study the idea in the context they used of chaos in a two dimensional map (which might be the Poincaré section of a three dimensional flow). Two reviews are [2] and [3].