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

TL;DR

This paper proves a conjecture that a finite number of time-delayed measurements can almost surely predict the future behavior of chaotic systems, extending previous results to non-invertible Lipschitz systems with arbitrary measures.

Contribution

It establishes the conjecture for all Lipschitz systems on compact sets with any Borel measure, and provides bounds on prediction accuracy decay rates.

Findings

Proves the conjecture for non-invertible Lipschitz systems.

Provides upper bounds for prediction error decay.

Extends time-delay prediction theorems to H"older systems.

Abstract

In the context of predicting the behaviour of chaotic systems, Schroer, Sauer, Ott and Yorke conjectured in 1998 that if a dynamical system defined by a smooth diffeomorphism $T$ of a Riemannian manifold $X$ admits an attractor with a natural measure $\mu$ of information dimension smaller than $k$, then $k$ time-delayed measurements of a one-dimensional observable $h$ are generically sufficient for $\mu$-almost sure prediction of future measurements of $h$. In a previous paper we established this conjecture in the setup of injective Lipschitz transformations $T$ of a compact set $X$ in Euclidean space with an ergodic $T$-invariant Borel probability measure $\mu$. In this paper we prove the conjecture for all (also non-invertible) Lipschitz systems on compact sets with an arbitrary Borel probability measure, and establish an upper bound for the decay rate of the measure of the set of points where the prediction is subpar. This partially confirms a second conjecture by Schroer, Sauer, Ott and Yorke related to empirical prediction algorithms as well as algorithms estimating the dimension and number of required delayed measurements (the so-called embedding dimension) of an observed system. We also prove general time-delay prediction theorems for locally Lipschitz or H\"older systems on Borel sets in Euclidean space.