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Prediction of dynamical systems from time-delayed measurements with self-intersections

Krzysztof Bara\'nskiYonatan GutmanAdam \'Spiewak
Dec 2022
摘要
In the context of predicting behaviour of chaotic systems, Schroer, Sauer,Ott and Yorke conjectured in 1998 that if a dynamical system defined by asmooth diffeomorphism $T$ of a Riemannian manifold $X$ admits an attractor witha natural measure $\mu$ of information dimension smaller than $k$, then $k$time-delayed measurements of a one-dimensional observable $h$ are genericallysufficient for $\mu$-almost sure prediction of future measurements of $h$. In aprevious paper we established this conjecture in the setup of injectiveLipschitz transformations $T$ of a compact set $X$ in Euclidean space with anergodic $T$-invariant Borel probability measure $\mu$. In this paper we provethe conjecture for all Lipschitz systems (also non-invertible) on compact setswith an arbitrary Borel probability measure, and establish an upper bound forthe decay rate of the measure of the set of points where the prediction issubpar. This partially confirms a second conjecture by Schroer, Sauer, Ott andYorke related to empirical prediction algorithms as well as algorithmsestimating the dimension and number of required delayed measurements (theso-called embedding dimension) of an observed system. We also prove generaltime-delay prediction theorems for locally Lipschitz or H\"older systems onBorel sets in Euclidean space.
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