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Unscented Transformation for Estimating the Lyapunov Exponents of Chaotic Time Series Corrupted by Random Noise

Authors: K. Kamalanand, P. Mannar Jawahar


Many systems in the natural world exhibit chaos or non-linear behavior, the complexity of which is so great that they appear to be random. Identification of chaos in experimental data is essential for characterizing the system and for analyzing the predictability of the data under analysis. The Lyapunov exponents provide a quantitative measure of the sensitivity to initial conditions and are the most useful dynamical diagnostic for chaotic systems. However, it is difficult to accurately estimate the Lyapunov exponents of chaotic signals which are corrupted by a random noise. In this work, a method for estimation of Lyapunov exponents from noisy time series using unscented transformation is proposed. The proposed methodology was validated using time series obtained from known chaotic maps. In this paper, the objective of the work, the proposed methodology and validation results are discussed in detail.

Keywords: Neural Networks, Chaos Theory, Lyapunov exponents, unscented transformation

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[1] N. L. Gerr, and J. C. Allen, “Stochastic versions of chaotic time series: generalized logistic and Henon time series models,” Physics D, vol. 68, 1993, pp. 232-249
[2] E. Takens, “Detecting nonlinearities in stationary time series,” International Journal of Bifurcation and Chaos, vol. 3, 1993, pp. 241- 256
[3] D. F. McCaffrey, S. Ellner, A. R. Gallant and D. W. Nychka, “Estimating the Lyapunov exponent of a chaotic system with nonparametric regression,” Journal of the Amer. Sta.l Assoc., vol. 87, 1992, pp. 682-695
[4] X. Zeng, R. Eykholt and R. A. Pielke, “Estimating the lyapunovexponent spectrum from short time series of low precision,” Phys. Rev. Let., vol. 66, 1991
[5] S. Mototsugu and L. Oliver, “Nonparametric neural network estimation of Lyapunov exponents and a direct test for chaos,” Journal of Econometrics, vol. 120, 2004, pp. 1–33
[6] A. N. Edmonds, “Time series prediction using supervised learning and tools from chaos theory,” Faculty of Science and Computing, University of Luton, Ph.D Thesis, 1996
[7] M. Ataei, A. Khaki-Sedigh, B. Lohmann and C. Lucas, “Estimating the lyapunov exponents of chaotic time series: a model based method,” Proceedings of European Control Conference, 2003
[8] H. Kantz, “A robust method to estimate the maximum Lyapunov exponent of a time series,” Phys. Lett. A, vol. 185, 1994, pp. 77-87
[9] M. Sano and Y. Sawada, “Measurement of the Lyapunov spectrum from a chaotic time series,” Phys. Rev. Let., vol. 55, 1985, pp. 1082-1085
[10] M. T. Rosenstein, J. J. Collins and C. J. De Luca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D, vol. 65, 1993, pp. 117-134
[11] A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D, vol. 16, 1985, pp. 285-317
[12] J.-P. Eckmann and D. Ruelle, “Ergodic theory of chaos and strange attractors,” Reviews of Modern Physics, vol. 57, 1985, pp. 617–656
[13] J.-P. Eckmann, S. O. Kamphorst, D. Ruelle and S. Ciliberto, “Liapunov exponents from time series,” Physical Review A, vol. 34, 1986, pp. 4971–4979
[14] D. Nychka, S. Ellner, A. R. Gallant and D. McCaHrey, “Finding chaos in noisy system,” Journal of the Royal Statistical Society, Series B, vol. 54, 1992, pp. 399–426
[15] S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proceedings of the IEEE, vol. 92, 2004, pp. 401-422
[16] Wan, E. and R. Van der Merwe, “The unscented Kalman filter,” Wiley Publishing, 2001
[17] Li Hongli, J. Wang, Y. Che, H. Wang and Y. Chen, “On neural network training algorithm based on the unscented Kalman filter,” Proceedings of 29th Chinese Control Conference, 2010, pp. 1447-1450
[18] J. C. Sprott, “Chaos and time series analysis,” oxford university press, 2003
[19] R. L. Devaney, “An introduction to chaotic dynamical systems,” 2nd edition, Addison Wesley, Redwood City, CA, 1989
[20] S. H. Strogatz, “Nonlinear dynamics and chaos,” Perseus publishing, 1994
[21] W. E. Ricker, “Stock and recruitment,” J. Fisheries Res. Board Can., vol. 11, 1954, pp. 559-623
[22] G. W. Frank, T. Lookman, M. A. H. Nerenberg, C. Essex, J. Lemieux and W. Blume, “Chaotic time series analysis of epileptic seizures,” Physica D, vol. 46, 1990, pp. 427
[23] P. Chen, “Empirical and theoretical evidence of economic chaos,” Sys. Dyn. Rev., vol. 4, 1988, pp. 81