The bootstrap and Lyapunov exponents in deterministic chaos

Inasmuch as Lyapunov exponents provide a necessary condition for chaos in a dynamical system, confidence bounds on estimated Lyapunov exponents are of great interest. Estimates derived either from observations or from numeric al integrations are limited to trajectories of finite length, and it is the uncertainties in (the distribution of) these finite time Lyapunov exponent s which are of interest. Within this context a bootstrap algorithm for quan tifying sampling uncertainties is shown to be inappropriate for multiplicat ive-ergodic statistics of deterministic chaos. This result remains unchange d in the presence of observational noise. As originally proposed, the algor ithm is also inappropriate for general nonlinear stochastic processes, a mo dified version is presented which may prove of value in the case of stochas tic dynamics. A new approach towards quantifying the minimum duration of ob servations required to estimate global Lyapunov exponents is suggested and is explored in a companion paper. (C) 1999 Elsevier Science B.V. All rights reserved.