Computational Statistics

, Volume 19, Issue 2, pp 159–168 | Cite as

Consistent Lyapunov exponent Estimation for one-dimensional dynamical systems

  • Salim LardjaneEmail author


The author proves the consistency of a nearest neighbor estimator of the Lyapunov exponent for a general class of one-dimensional ergodic dynamical systems. The author shows that this estimator has good practical properties on a set of simulations.


Dynamical Systems Chaos Ergodicity Lyapunov Exponent Nearest Neighbors Estimation 


  1. Boyarsky, A. &Góra, P. (1997),Laws of Chaos, Probability and its applications, Birkhaüser.Google Scholar
  2. Delecroix, M., Guegan, D. &Léorat, G. (1997), ’Determinating Lyapunov exponents in deterministic dynamical systems’,Computational Statistics 12, 93–107.MathSciNetzbMATHGoogle Scholar
  3. Eckmann, J., Kamphorst, S., Ruelle, D. &Ciliberto, S. (1986), ’Lyapunov exponents from time series’,Physical Review A 34(6), 4971–4979.MathSciNetCrossRefGoogle Scholar
  4. Ellner, S., Gallant, A., McCaffrey, D. &Nychka, D. (1991), ’Convergence rates and data requirements for jacobian-based estimates of Lyapunov exponents from data’,Physics Letters A 153(6, 7), 357–363.MathSciNetCrossRefGoogle Scholar
  5. Embrechts, M. (1994), Basic concepts of nonlinear dynamics and chaos theory,in G. Deboeck, ed., ’Trading on the edge: Neural, Genetic, and Fuzzy Systems for Chaotic Financial Markets’, Wiley.Google Scholar
  6. Holzfuss, J. &Parlitz, U. (1990), Lyapunov exponents from time series,in L. Arnold &J.-P. Eckmann, eds, ’Lyapunov Exponents: Proceedings, Oberwolfach 1990’, number 1486in ’Lecture Notes in Mathematics’, Springer-Verlag, pp. 263–270.zbMATHGoogle Scholar
  7. May, R. (1976), ’Simple mathematical models with very complicated dynamics’,Nature 261, 459–467.CrossRefGoogle Scholar
  8. McCaffrey, D., Ellner, S., Gallant, A. &Nychka, D. (1992), ’Estimating the lyapunov exponent of a chaotic system with nonparametric regression’,Journal of the American Statistical Association 87, 682–695.MathSciNetCrossRefGoogle Scholar
  9. Medio, A. (1992),Chaotic Dynamics: Theory and applications to Economics, Cambridge.Google Scholar
  10. Neimark, Y. &Landa, P. (1992),Stochastic and Chaotic Oscillations, Kluwer Academic Publishers.Google Scholar
  11. Nychka, D., Ellner, S., Gallant, A. &McCaffrey, D. (1992), ’Finding chaos in noisy systems’,Journal of the Royal Statistical Society, Series B 54, 399–426.MathSciNetGoogle Scholar
  12. Vastano, J. &Kostelich, D. (1986), Comparison of algorithms for determining Lyapunov exponents from experimental data,in G. Mayer-Kress, ed., ’Dimension and Entropies in Chaotic Systems: Quantification of Complex Behavior’, Springer-Verlag.CrossRefGoogle Scholar
  13. Whang, Y.-J. &Linton, O. (1999), ’The asymptotic distribution of nonparametric estimates of the lyapunov exponent for stochastic time series’,Journal of Econometrics 91, 1–42.MathSciNetCrossRefGoogle Scholar
  14. Wolff, A., Swift, J., Swinney, H. &Vastano, J. (1985), ’Determining Lyapunov exponents from a time series’,Physica D (16), 285–317.MathSciNetCrossRefGoogle Scholar

Copyright information

© Physica-Verlag 2004

Authors and Affiliations

  1. 1.Laboratoire de Statistique et Modélisation ENSAIUniversité de Bretagne Sud & CREST CRESTBruzFrance

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