Bursts in the Chaotic Trajectory Lifetimes Preceding the Controlled Periodic Motion

Abstract

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