Control of chaos by occasional bang-bang

The stabilization of a periodic saddle orbit (the target orbit) in a strange attractor is usually achieved by the application of a sequence of parameter perturbations designed to place the system state in the stable manifold of the target, whereupon it evolves unperturbed to the target orbit. Controls of this type originated with the method of Ott, Grebogi, and Yorke (OGY), and usually require a continuously variable parameter for map based control. Bang-bang control is a method whereby control is achieved by the application of a fixed, or several different fixed perturbations, rather than a continuous range of perturbations, and generally requires a flexible scheduling. If we have available only fixed perturbation levels, and can apply control only at regular intervals and for fixed durations, standard OGY will not work. We demonstrate a method that will control maps and continuous systems at a surface of section, albeit imprecisely, with a single fixed perturbation of fixed duration. We call this occasional bang bang.