Controlling chaos experimentally in systems exhibiting large effective Lyapunov exponents

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Abstract
We investigate experimentally the performance of the Ott, Grebogi, and Yorke [Phys. Rev. Lett. 64, 1196 (1989)] feedback concept to control chaotic motion. The experimental systems are a driven pendulum and a driven bronze ribbon. Both setups have unstable periodic orbits characterized by large effective Lyapunov exponents. All control vectors for the feedback control are extracted from the experimental data. To do this for the pendulum a global model obtained by the flow field analysis of Cremers and Hübler [Z. Naturforsch. 42a, 797 (1987)] is used, and for the bronze ribbon linear approximations in embedding space are exploited. We analyze the problems that arise due to the amplification of noise by large effective Lyapunov exponents in the determination of the control values as well as in the performance of the experimental control. Successful control can be achieved in our experiments by applying the ‘‘local control method’’ which allows a quasicontinuous adjustment of the control parameter in contrast to adjusting the control parameter only once per return time of the Poincaré map.

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