Nonlinear prediction as a way of distinguishing chaos from random fractal sequences

Abstract

NONLINEAR forecasting has recently been shown to distinguish between deterministic chaos and uncorrelated (white) noise added to periodic signals¹, and can be used to estimate the degree of chaos in the underlying dynamical system². Distinguishing the more general class of coloured (autocorrelated) noise has proven more difficult because, unlike additive noise, the correlation between predicted and actual values measured may decrease with time—a property synonymous with chaos. Here, we show that by determining the scaling properties of the prediction error as a function of time, we can use nonlinear prediction to distinguish between chaos and random fractal sequences. Random fractal sequences are a particular class of coloured noise which represent stochastic (infinite-dimensional) systems with power-law spectra. Such sequences have been known to fool other procedures for identifying chaotic behaviour in natural time series⁹, particularly when the data sets are small. The recognition of this type of noise is of practical importance, as measurements from a variety of dynamical systems (such as three-dimensional turbulence, two-dimensional and geostrophic turbulence, internal ocean waves, sandpile models, drifter trajectories in large-scale flows, the motion of a classical electron in a crystal and other low-dimensional systems) may over some range of frequencies exhibit power-law spectra.