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Consistent Lyapunov exponent Estimation for one-dimensional dynamical systems

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Summary

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.

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References

  • Boyarsky, A. &Góra, P. (1997),Laws of Chaos, Probability and its applications, Birkhaüser.

  • Delecroix, M., Guegan, D. &Léorat, G. (1997), ’Determinating Lyapunov exponents in deterministic dynamical systems’,Computational Statistics 12, 93–107.

    MathSciNet  MATH  Google Scholar 

  • Eckmann, J., Kamphorst, S., Ruelle, D. &Ciliberto, S. (1986), ’Lyapunov exponents from time series’,Physical Review A 34(6), 4971–4979.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

  • 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.

    MATH  Google Scholar 

  • May, R. (1976), ’Simple mathematical models with very complicated dynamics’,Nature 261, 459–467.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Medio, A. (1992),Chaotic Dynamics: Theory and applications to Economics, Cambridge.

  • Neimark, Y. &Landa, P. (1992),Stochastic and Chaotic Oscillations, Kluwer Academic Publishers.

  • 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.

    MathSciNet  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Wolff, A., Swift, J., Swinney, H. &Vastano, J. (1985), ’Determining Lyapunov exponents from a time series’,Physica D (16), 285–317.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Statistique et Modélisation ENSAI, Université de Bretagne Sud & CREST CREST, Rue Blaise Pascal, 35170, Bruz, France

    Salim Lardjane

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  1. Salim Lardjane
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Correspondence to Salim Lardjane.

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Lardjane, S. Consistent Lyapunov exponent Estimation for one-dimensional dynamical systems. Computational Statistics 19, 159–168 (2004). https://doi.org/10.1007/BF02892054

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