ANALYSIS OF THE CONTROL OF CHAOS - EXTENDING THE BASIN OF ATTRACTION

The method of controlling chaos using small parameter perturbations wh ich 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 maybe small if the maximum pe rmissible parameter perturbation is small, resulting in long chaotic t ransients 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 par ameter perturbation may be made before attempting to place an iterate on the stable manifold of the fixed point using an appropriately chose n perturbation of the parameter. This has the effect of reducing trans ient times by a factor of approximately \lambda(u)\/(\lambda(u)\ -1) w here lambda(u) is the unstable eigenvalue of the saddle fixed point, a s 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 nume rical example. (C) 1997 Elsevier Science Ltd.