2024/01/28 by Krzysztof Barański, Barański, Krzysztof, Yonatan Gutman +3 · 1 citation
Engineering · #37C40 #37C45 #58D10 #Advanced Control Systems Optimization #Control Systems and Identification #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Fault Detection and Control Systems #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2401.15712
openalex publication_date 2024/01/28 · openalex created_date 2024/01/31 · openalex updated_date 2026/07/28
We study the problem of reconstructing and predicting the future of a dynamical system by the use of time-delay measurements of typical observables. Considering the case of too few measurements, we prove that for Lipschitz systems on compact sets in Euclidean spaces, equipped with an invariant Borel probability measure μ of Hausdorff dimension d, one needs at least d measurements of a typical (prevalent) Lipschitz observable for μ-almost sure reconstruction and prediction. Consequently, the Hausdorff dimension of μ is the precise threshold for the minimal delay (embedding) dimension for such systems in a probabilistic setting. Furthermore, we establish a lower bound postulated in the Schroer--Sauer--Ott--Yorke prediction error conjecture from 1998, after necessary modifications (whereas the upper estimates were obtained in our previous work). To this aim, we prove a general theorem on the dimensions of conditional measures of μ with respect to time-delay coordinate maps.