Analysis of the control of chaos—Extending the basin of attraction

The method of controlling chaos using small parameter perturbations which was first proposed by Ott, Grebogi and Yorke (Phys. Rev. Lett., 64, 1196–1199) suffers from two problems: the ‘basin of attraction’ for the fixed point which will be stabilized may be small if the maximum permissible parameter perturbation is small, resulting in long chaotic transients before control is achieved, and noise can result in control being lost. We address both these problems by constructing an extended basin of attraction in which several iterations using the maximum parameter perturbation may be made before attempting to place an iterate on the stable manifold of the fixed point using an appropriately chosen perturbation of the parameter. This has the effect of reducing transient times by a factor of approximately where λu is the unstable eigenvalue of the saddle fixed point, as well as reducing the effects of noise. The method is also applied to other related control methods and it is shown that the same extended basin of attraction is obtained. The method is illustrated with a numerical example.