Transient Behavior for One-Dimensional ogy Control

The control process for a chaotic system as proposed by Ott, Grebogi and Yorke (OGY) is investigated for systems with a one-dimensional Poincaré map. The calculation of the Poinaré map for the controlled system turns out to be an instructive method to analyze the mechanism of OGY control. We introduce the notion of a region of immediate capturing, characterizing the maximum control region that guarantees success of OGY control. Concerning the mean time to achieve control, we show that the simple probability argument in OGY’s original paper has to be refined. The exponent of OGY’s asymptotic power law is not affected by this detail, but the scale factor is incorrect. For the case of one-dimensional maps we derive a modified law for the dependence of the mean time to achieve control on the size of the control region. In contrast to OGY’s prediction this formula is also valid for large control bounds and fits well with numerical calculations. The revised statistical argument is corroborated by investigating the time evolution of the density during control. This approach allows an alternative determination of the mean time to achieve control. The evolution of a stationary density shape is demonstrated, which notably differs from the invariant density even for moderate control bounds. This observation explains minor deviations from the modified law that can be found especially for large control bounds.