DISCRETE-TIME DYNAMICAL-SYSTEMS UNDER OBSERVATIONAL UNCERTAINTY

Discrete-time dynamical systems under observational uncertainty are st udied. As a result of the uncertainty, points on an orbit are surround ed by uncertainty sets. The problem of reconstructing the original orb it given the sequence of uncertainty sets is investigated. The key pro perty which makes the reconstruction possible is the sensitivity to in itial conditions. A general reconstructing algorithm is theoretically analyzed and experimentally tested on several low-dimensional systems. The technique is extended to coupled one-dimensional maps with the go al of eventually developing retrospective techniques for partial diffe rential equations exhibiting spatio-temporal chaos. Provided the coupl ing strength remains small and the coupling term has bounded first der ivatives, it is conjectured that for dynamical systems with a positive Liapunov exponent the observational uncertainty can be reduced expone ntially with the length of the orbit used for reconstruction. Computer experiments with the coupled logistic map are consistent with this co njecture. (C) Elsevier Science Inc., 1997.