Measurement of the Lyapunov Spectrum from a Chaotic Time Series

A new method is proposed to determine the spectrum of several Lyapunov exponents (including positive, zero, and even negative ones) from the observed time series of a single variable.

Abstract

The exponential divergence or convergence of nearby trajectories (Lyapunov exponents) is conceptually the most basic indicator of deterministic chaos. We propose a new method to determine the spectrum of several Lyapunov exponents (including positive, zero, and even negative ones) from the observed time series of a single variable. We have applied the method to various known model systems and also to the Rayleigh-B\'enard experiment, and have elucidated the dependence of the Lyapunov exponents on the Rayleigh number.