HISTORY-DEPENDENT CONTROL OF UNSTABLE PERIODIC-ORBITS

We present a method of stabilizing unstable periodic orbits in systems whose natural time scales are on the order of or faster than the time it would take for the experimental implementation of the Ott-Grebogi- Yorke (OGY) controlling method [Phys. Rev. Lett. 84, 1196 (1990)]. We determine the controlling perturbation one or more cycles ahead of whe n it needs to be applied, thereby gaining the additional time necessar y to measure a signal, determine the perturbation, and then implement it. Formulas for this method of prior iterate control are derived and their utility is demonstrated numerically on the Henon map for control ling the unstable orbits of period one and two, The effects of noise o n this control method are examined and the results are compared with a similar application of the OGY scheme in the presence of noise.