Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2018, Article ID 6909151, 8 pages
https://doi.org/10.1155/2018/6909151
Research Article

On the Use of Interval Extensions to Estimate the Largest Lyapunov Exponent from Chaotic Data

1Control and Modelling Group (GCOM), Department of Electrical Engineering, Federal University of São João del-Rei, 36307-352 São João del-Rei, MG, Brazil
2Department of Electronic Engineering, School of Engineering, Federal University of Minas Gerais, 31270-901 Belo Horizonte, MG, Brazil

Correspondence should be addressed to Márcio J. Lacerda; rb.ude.jsfu@adrecal

Received 28 November 2017; Revised 12 January 2018; Accepted 28 January 2018; Published 4 March 2018

Academic Editor: Gisele Mophou

Copyright © 2018 Erivelton G. Nepomuceno et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. 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: Nonlinear Phenomena, vol. 65, no. 1-2, pp. 117–134, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. V. I. Oseledec, “The multiplicative ergodic theorem: the lyapunov characteristic numbers of dynamical systems,” Transactions of the Moscow Mathematical Society, vol. 19, pp. 179–210, 1968. View at Google Scholar · View at MathSciNet
  3. G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, “Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application,” Meccanica, vol. 15, no. 1, pp. 21–30, 1980. View at Publisher · View at Google Scholar · View at Scopus
  4. G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: theory,” Meccanica, vol. 15, no. 1, pp. 9–20, 1980. View at Publisher · View at Google Scholar · View at Scopus
  5. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D: Nonlinear Phenomena, vol. 16, no. 3, pp. 285–317, 1985. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. M. Sano and Y. Sawada, “Measurement of the Lyapunov spectrum from a chaotic time series,” Physical Review Letters, vol. 55, no. 10, pp. 1082–1085, 1985. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. H. Kantz, “A robust method to estimate the maximal Lyapunov exponent of a time series,” Physics Letters A, vol. 185, no. 1, pp. 77–87, 1994. View at Publisher · View at Google Scholar · View at Scopus
  8. G. Rangarajan, S. Habib, and R. D. Ryne, “Lyapunov exponents without rescaling and reorthogonalization,” Physical Review Letters, vol. 80, no. 17, pp. 3747–3750, 1998. View at Publisher · View at Google Scholar · View at Scopus
  9. K. Geist, U. Parlitz, and W. Lauterborn, “Comparison of different methods for computing Lyapunov exponents,” Progress of Theoretical and Experimental Physics, vol. 83, no. 5, pp. 875–893, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  10. B. J. Kim and G. H. Choe, “High precision numerical estimation of the largest Lyapunov exponent,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 5, pp. 1378–1384, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. G. Wainrib and M. N. Galtier, “A local Echo State Property through the largest Lyapunov exponent,” Neural Networks, vol. 76, pp. 39–45, 2016. View at Publisher · View at Google Scholar · View at Scopus
  12. A. J. Steyer and E. S. Van Vleck, “A step-size selection strategy for explicit Runge-Kutta methods based on Lyapunov exponent theory,” Journal of Computational and Applied Mathematics, vol. 292, pp. 703–719, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. C. J. Gavilán-Moreno and G. Espinosa-Paredes, “Using Largest Lyapunov Exponent to Confirm the Intrinsic Stability of Boiling Water Reactors,” Nuclear Engineering and Technology, vol. 48, no. 2, pp. 434–447, 2016. View at Publisher · View at Google Scholar · View at Scopus
  14. W. Caesarendra, B. Kosasih, A. K. Tieu, and C. A. S. Moodie, “Application of the largest Lyapunov exponent algorithm for feature extraction in low speed slew bearing condition monitoring,” Mechanical Systems and Signal Processing, vol. 50-51, pp. 116–138, 2015. View at Publisher · View at Google Scholar · View at Scopus
  15. E. M. A. M. Mendes and E. G. Nepomuceno, “A very simple method to calculate the (Positive) largest lyapunov exponent using interval extensions,” International Journal of Bifurcation and Chaos, vol. 26, no. 13, Article ID 1650226, 2016. View at Publisher · View at Google Scholar · View at Scopus
  16. E. G. Nepomuceno and S. A. M. Martins, “A lower bound error for free-run simulation of the polynomial NARMAX,” Systems Science & Control Engineering, vol. 4, no. 1, pp. 50–58, 2016. View at Publisher · View at Google Scholar · View at Scopus
  17. E. Nepomuceno, S. Martins, G. Amaral, and R. Riveret, “On the lower bound error for discrete maps using associative property,” Systems Science Control Engineering, vol. 5, no. 1, pp. 462–473, 2017. View at Publisher · View at Google Scholar
  18. S. A. Billings, Nonlinear system identification: NARMAX methods in the time, frequency, and spatio-temporal domains, West Sussex: John Wiley & Sons, London, UK, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  19. L. A. Aguirre and C. Letellier, “Modeling nonlinear dynamics and chaos: A review,” Mathematical Problems in Engineering, vol. 2009, Article ID 238960, 2009. View at Publisher · View at Google Scholar · View at Scopus
  20. L. A. Aguirre and S. Billings, “Retrieving dynamical invariants from chaotic data using narmax models,” International Journal of Bifurcation and Chaos, vol. 05, no. 02, pp. 449–474, 1995. View at Publisher · View at Google Scholar
  21. J. C. Sprott, “A new chaotic jerk circuit,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 58, no. 4, pp. 240–243, 2011. View at Publisher · View at Google Scholar · View at Scopus
  22. S. Chen and S. A. Billings, “Representations of nonlinear systems: the {NARMAX} model,” International Journal of Control, vol. 49, no. 3, pp. 1013–1032, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. L. Piroddi and W. Spinelli, “An identification algorithm for polynomial {NARX} models based on simulation error minimization,” International Journal of Control, vol. 76, no. 17, pp. 1767–1781, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. J. R. Ayala Solares, H.-L. Wei, and S. A. Billings, “A novel logistic-NARX model as a classifier for dynamic binary classification,” Neural Computing and Applications, pp. 1–15, 2017. View at Publisher · View at Google Scholar · View at Scopus
  25. O. M. Mohamed Vall and R. M'hiri, “An approach to polynomial NARX/NARMAX systems identification in a closed-loop with variable structure control,” International Journal of Automation and Computing, vol. 5, no. 3, pp. 313–318, 2008. View at Publisher · View at Google Scholar · View at Scopus
  26. E. M. Mendes and S. A. Billings, “On Identifying Global Nonlinear Discrete Models from Chaotic Data,” International Journal of Bifurcation and Chaos, vol. 07, no. 11, pp. 2593–2601, 1997. View at Publisher · View at Google Scholar
  27. L. A. Aguirre, G. G. Rodrigues, and E. M. Mendes, “Nonlinear Identification and Cluster Analysis of Chaotic Attractors from a Real Implementation of Chua's Circuit,” International Journal of Bifurcation and Chaos, vol. 07, no. 06, pp. 1411–1423, 1997. View at Publisher · View at Google Scholar
  28. W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, 3rd edition, 1976. View at MathSciNet
  29. R. Gilmore, M. Lefranc, and N. B. Tufillaro, The topology of chaos: Alice in stretch and squeezeland, Weinheim Chichester: Wiley-VCH John Wiley distributor, 2011. View at Publisher · View at Google Scholar · View at Scopus
  30. M. L. Overton, Numerical computing with {IEEE} floating point arithmetic, Society for Industrial and Applied Mathematics, 2001. View at MathSciNet
  31. IEEE Standards Committee and others, “754-2008 IEEE standard for floating-point arithmetic,” IEEE Computer Society Std, 2008. View at Publisher · View at Google Scholar
  32. E. G. Nepomuceno and E. M. A. M. Mendes, “On the analysis of pseudo-orbits of continuous chaotic nonlinear systems simulated using discretization schemes in a digital computer,” Chaos, Solitons & Fractals, vol. 95, pp. 21–32, 2017. View at Publisher · View at Google Scholar · View at Scopus
  33. M. V. Correa, L. A. Aguirre, and E. M. A. M. Mendes, “Modeling chaotic dynamics with discrete nonlinear rational models,” International Journal of Bifurcation and Chaos, vol. 10, pp. 1019–1032, 2000. View at Google Scholar
  34. E. G. Nepomuceno, R. H. Takahashi, G. F. Amaral, and L. A. Aguirre, “Nonlinear identification using prior knowledge of fixed points: a multiobjective approach,” International Journal of Bifurcation and Chaos, vol. 13, no. 5, pp. 1229–1246, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  35. O. E. Rössler, “An equation for continuous chaos,” Physics Letters A, vol. 57, no. 5, pp. 397-398, 1976. View at Publisher · View at Google Scholar · View at Scopus
  36. M. C. Mackey and L. Glass, “Oscillation and chaos in physiological control systems,” Science, vol. 197, no. 4300, pp. 287–289, 1977. View at Publisher · View at Google Scholar · View at Scopus
  37. J. Doyne Farmer, “Chaotic attractors of an infinite-dimensional dynamical system,” Physica D: Nonlinear Phenomena, vol. 4, no. 3, pp. 366–393, 1982. View at Publisher · View at Google Scholar · View at Scopus
  38. X. Zeng, R. Eykholt, and R. A. Pielke, “Estimating the Lyapunov-exponent spectrum from short time series of low precision,” Physical Review Letters, vol. 66, no. 25, pp. 3229–3232, 1991. View at Publisher · View at Google Scholar · View at Scopus