Bursts in the Chaotic Trajectory Lifetimes Preceding the Controlled Periodic Motion

The average lifetime ($\tau (H)$) it takes for a randomly started trajectory to land in a small region ($H$) on a chaotic attractor is studied. $\tau (H)$ is an important issue for controlling chaos. We point out that if the region $H$ is visited by a short periodic orbit, the lifetime $\tau (H)$ strongly deviates from the inverse of the naturally invariant measure contained within that region (${\mu }_N(H{)}^{-1}$). We introduce the formula that relates $\tau (H)\mathrm{/}{\mu }_N(H{)}^{-1}$ to the expanding eigenvalue of the short periodic orbit visiting $H$.