Direct dynamical test for deterministic chaos and optimal embedding of a chaotic time series

Abstract

We propose here a local exponential divergence plot which is capable of providing an alternative means of characterizing a complex time series. The suggested plot defines a time-dependent exponent and a ‘‘plus’’ exponent. Based on their changes with the embedding dimension and delay time, a criterion for estimating simultaneously the minimal acceptable embedding dimension, the proper delay time, and the largest Lyapunov exponent has been obtained. When redefining the time-dependent exponent Λ(k) curves on a series of shells, we have found that whether a linear envelope to the Λ(k) curves exists can serve as a direct dynamical method of distinguishing chaos from noise.