Controlling chaos using time delay coordinates.

This chapter discusses controlling chaos using time delay coordinates. The Ott-Grebogi-Yorke (OGY) control method is analyzed in the case that the attractor is reconstructed from a time series using time delay coordinates. It turns out that the control formula of Ott, Grebogi and Yorke should be modified in order to apply to experimental systems if time delay coordinates are used. The chapter reveals that the experimental surface of section map depends not only on the actual parameter but also on the preceding one. In order to meet this dependence two modifications are introduced which lead to a better performance of the control. To compare their control abilities they are applied to simulations of a Duffing oscillator. OGY proposed a new method of controlling a chaotic dynamical system by stabilizing one of the many unstable periodic orbits embedded in a chaotic attractor, through only small time dependent perturbations in some accessible system parameter. This makes OGY's approach quite different from other previously published methods on controlling chaos. OGY's method has attracted the attention of many physicists interested in applications of nonlinear dynamics.