Anomalous convergence of Lyapunov exponent estimates
- Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)
- 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, Journal Name: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics Journal Issue: 4 Vol. 51; ISSN PLEEE8; ISSN 1063-651X
- Country of Publication:
- United States
- Language:
- English
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