The Cramer-Rao bound for initial conditions estimation of chaotic orbits

We derive the Cramer-Rao Lower Bound (CRLB) for the estimation of initial conditions of noise-embedded orbits produced by general one-dimensional maps. We relate this bound’s asymptotic behavior to the attractor’s Lyapunov number and show numerical examples. These results pave the way for more suitable choices for the chaotic signal generator in some chaotic digital communication systems.