Controlling chaos using nonlinear approximations and delay coordinate embedding

In a previous paper we showed that a chaos control method proposed by Ott, Grebogi and Yorke can be improved by using nonlinear approximations for chaotic dynamical systems and stable manifolds of targets. Here we consider systems whose governing equations are unknown and apply the chaos control method using the nonlinear approximations. Delay coordinate embedding techniques are used, so that approximate saddle points to be stabilized and nonlinear approximations of the systems and stable manifolds are obtained from time series of single variables. We also take into account the fact that the obtained section maps depend on the current and previous parameters. To demonstrate our approach, we give two numerical examples for the Henon map and a pendulum with feedforward and feedback control. Some influences of noise are also discussed in these examples. (C) 1998 Elsevier Science B.V.