Anomalous convergence of Lyapunov exponent estimates

Journal Article · Sat Apr 01 00:00:00 EST 1995 · Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

DOI:https://doi.org/10.1103/PhysRevE.51.3738· OSTI ID:27855

Theiler, J ^[1]; Smith, L A ^[2]

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1. Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)
2. Mathematical Institute, University of Oxford, Oxford, OX13LB (United Kingdom)

Numerical experiments reveal that estimates of the Lyapunov exponent for the logistic map {ital x}{sub {ital t}+1}={ital f}({ital x}{sub {ital t}})=4{ital x}{sub {ital t}}(1{minus}{ital x}{sub {ital t}}) are anomalously precise: they are distributed with a standard deviation that scales as 1/{ital N}, where {ital N} is the length of the trajectory, not as 1/ {radical}{ital N} , the scaling expected from an informal interpretation of the central limit theorem. We show that this anomalous convergence follows from the fact that the logistic map is conjugate to a constant-slope map. The Lyapunov estimator is just one example of a ``chaotic walk``; we show that whether or not a general chaotic walk exhibits anomalously small variance depends only on the autocorrelation of the chaotic process.

• OSTI ID: 27855
• Journal Information: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, Vol. 51, Issue 4; Other Information: PBD: Apr 1995
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
LYAPUNOV METHOD
CONVERGENCE
TIME-SERIES ANALYSIS
MAPPING
SCALING
PROBABILISTIC ESTIMATION
ERRORS
CORRELATIONS