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Studies in Nonlinear Dynamics & Econometrics

Ed. by Mizrach, Bruce

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Volume 1, Issue 3


The Identification of Spurious Lyapunov Exponents in Jacobian Algorithms

Ramazan Gencay
  • Department of Economics, Department of Mathematics and Statistics, University of Windsor, Windsor, Ontario, Canada, gencay@uwin
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ W. Davis Dechert
Published Online: 1996-10-01 | DOI: https://doi.org/10.2202/1558-3708.1018

The method of reconstructing an n-dimensional system from observations is to form vectors of m consecutive observations, which for m 2n, is generically an embedding. This is Takens's result. The Jacobian methods for Lyapunov exponents utilize a function of m variables to model the data, and the Jacobian matrix is constructed at each point in the orbit of the data. When embedding occurs at dimension m = n, the Lyapunov exponents of the reconstructed dynamics are the Lyapunov exponents of the original dynamics. However, if embedding only occurs for an m > n, then the Jacobian method yields m Lyapunov exponents, only n of which are the Lyapunov exponents of the original system. The problem is that as currently used, the Jacobian method is applied to the full m-dimensional space of the reconstruction, and not just to the n-dimensional manifold that is the image of the embedding map. Our examples show that it is possible to obtain spurious Lyapunov exponents that are even larger than the largest Lyapunov exponent of the original system.

This article offers supplementary material which is provided at the end of the article.

Keywords: Lyapunov exponents; embedded dynamics

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Published Online: 1996-10-01

Citation Information: Studies in Nonlinear Dynamics & Econometrics, Volume 1, Issue 3, ISSN (Online) 1558-3708, DOI: https://doi.org/10.2202/1558-3708.1018.

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©2011 Walter de Gruyter GmbH & Co. KG, Berlin/Boston. Copyright Clearance Center

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