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# A practical method for calculating Lyapunov exponents from small data sets

Published: Jan. 16, 2001. Version: 1.0.0

When using this resource, please cite the original publication:

M.T. Rosenstein, J.J. Collins, and C.J. De Luca. A practical method for calculating largest Lyapunov exponents from small data sets. This article originally appeared in Physica D 65:117-134, 1993

Please include the standard citation for PhysioNet:

Goldberger AL, Amaral LAN, Glass L, Hausdorff JM, Ivanov PCh, Mark RG, Mietus JE, Moody GB, Peng C-K, Stanley HE. PhysioBank, PhysioToolkit, and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals (2003). Circulation. 101(23):e215-e220.

### Abstract

Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and estimate the amount of chaos in a system. We present a new method for calculating the largest Lyapunov exponent from an experimental time series. The method follows directly from the definition of the largest Lyapunov exponent and is accurate because it takes advantage of all the available data. We show that the algorithm is fast, easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level. Furthermore, one may use the algorithm to calculate simultaneously the correlation dimension. Thus, one sequence of computations will yield an estimate of both the level of chaos and the system complexity.

The full article may be downloaded in PDF (783KB) or gzip-compressed PostScript (361KB) formats.

Visualizing the effects of filtering chaotic signals

Reconstruction expansion as a geometry-based framework for choosing proper delay times

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Topics:
chaos complexity

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