from PhysioNet, the research resource for complex physiologic signals

A practical method for calculating Lyapunov exponents from small data sets

[Return to PhysioToolkit]

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 cite this publication when referencing this material. Software that implements the algorithm described by this article may be found here.


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.

See also

Visualizing the effects of filtering chaotic signals

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

Send feedback about this page to PhysioNet

Your comments and suggestions are welcome. We encourage you to use our feedback form to comment on this page. If you would like to receive a reply, please send your comments by email to

Updated Thursday, 9 July 2015 at 11:24 EDT

NIBIB/NIH/NIGMS National Institute for Biomedical Imaging and Bioengineering National Institutes of Health National Institute of General Medical Sciences