Quantifying Chaos with Predictive Flows and Maps: Locating Unstable Periodic Orbits

Abstract

Several authors have suggested methods for constructing "predictors" from time series data. It is shown that using one type of predictor, constructed with radial basis functions, for some known chaotic systems, the existence of unstable periodic orbits may be established using much less data than that required by alternative methods. The general question of quantifying the error in a predictor is also addressed. Considering the fraction of the data that can be predicted as a function of the accuracy of the prediction provides a method of distinguishing different sources of error in the predictor and, in doing so, yields an estimate of the magnitude and distribution of the observational noise in the system.