# Discrete-Time Dynamical Systems under Observational Uncertainty

## Abstract

Discrete-time dynamical systems under observational uncertainty are studied. As a result of the uncertainty, points on an orbit are surrounded by uncertainty sets. The problem of reconstructing the original orbit given the sequence of uncertainty sets is investigated. The key property which makes the reconstruction possible is the sensitivity to initial conditions. A general reconstructing algorithm is theoretically analysed and experimentally tested on several low-dimensional systems. The technique is extended to coupled one-dimensional maps with the goal of eventually developing retrospective techniques for partial differential equations exhibiting spatio-temporal chaos. Provided the coupling strength remains small and the coupling term has bounded first derivatives, it is conjectured that for dynamical systems with a positive Lyapunov exponent the observational uncertainty can be reduced exponentially with the length of the orbit used for reconstruction. Computer experiments with the coupled logistic map are consistent with this conjecture.

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