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An Optimization Approach for Finding a Spectrum of Lyapunov Exponents

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Computational Neuroscience

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 38))

Abstract

In this chapter, we consider an optimization technique for estimating the Lyapunov exponents from nonlinear chaotic systems. We then describe an algorithm for solving the optimization model and discuss the computational aspects of the proposed algorithm. To show the efficiency of the algorithm, we apply it to some well-known data sets. Numerical tests show that the algorithm is robust and quite effective, and its performance is comparable with that of other well-known algorithms.

This research is supported by NSF and Air Force grants.

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Notes

  1. 1.

    See Oseledec's theorem in [12].

  2. 2.

    In the implementation, among the N displacement vectors found inside the sphere of radius ε, only five to seven vectors with the smallest norm are chosen. N is often chosen as \(d_E \leq N \leq 20\) [28] and is kept at a low value to optimize the efficiency of the algorithm.

  3. 3.

    Throughout this chapter 512 bins have been used.

  4. 4.

    \(1\ {\textrm{nat}}/{\textrm{s}}\approx 1.44 {\textrm{bits}}/{\textrm{s}}\).

  5. 5.

    We have computed Equation (16.16) between 30 and 40 times per mean orbit.

References

  1. Abarbanel, H., Brown, R., Kennel, M. Variation of Lyapunov exponents in chaotic systems: Their importance and their evaluation using observed data. J Nonlinear Sci 2, 343–365 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, G., Lai, D. Making a dynamical system chaotic: Feedback control of Lyapunov exponents for discrete-time dynamical systems. IEEE Trans Circuits Syst I Fundam Theory Appl 44, 250–253 (1997)

    Article  MathSciNet  Google Scholar 

  3. Cvitanovic, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G. Classical and Quantum Chaos. 10th ed., ChaosBook.org Niels Bohr Institute, Copenhagen (2003) Webbook: http://chaosbook.org/.

  4. Djamai, L., Coirault, P. Estimation of Lyapunov exponents by using the perceptron. Proceedings of the American Control Conference 6, 5150–5155 (2002)

    Google Scholar 

  5. Eckmann, J.P., Kamphorst, S.O., Ruelle, D., Ciliberto, S. Lyapunov exponents from time series. Phys Rev A 34, 4971–4979 (1986)

    Article  MathSciNet  Google Scholar 

  6. Eckmann, J.P., Ruelle, D. Ergodic theory of chaos and strange attractors. Rev Modern Phys 57, 617–657 (1985)

    Article  MathSciNet  Google Scholar 

  7. Elger, C.E., Lehnertz, K. Seizure prediction by non-linear time series analysis of brain electrical activity. Eur J Neurosci 10, 786–789 (1998)

    Article  Google Scholar 

  8. Golovko, V. Estimation of the Lyapunov spectrum from one-dimensional observations using neural networks. Proceedings of the Second IEEE International Workshop on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications, 95–98 (2003)

    Google Scholar 

  9. Hilborn, R.C. Chaos and Nonlinear Dynamics. Oxford University Press, New York (2000)

    Book  Google Scholar 

  10. Iasemidis, L.D., Pardalos, P.M., Sackellares, J.C., Yatsenko, V.A. Global optimization approaches to reconstruction of dynamical systems related to epileptic seizures. In: Fotiadis, D., Massalas, C.V. (eds.) Scattering and Biomedical Engineering: Modeling and Applications, World Scientific, Singapore, pp. 308–318 (2002)

    Google Scholar 

  11. Kantz, H., Schreiber, T. Nonlinear Time Series Analysis. Cambridge University Press, New York (1997).

    Google Scholar 

  12. Kelliher, J.: Lyapunov Exponents and Oseledec's Multiplicative Ergodic Theorem. http://www.ma.utexas.edu/users/kelliher/Geometry/Geometry.html (2005)

  13. Kennel, M.B., Brown, R., Abarbanel, H.D.I. Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys Rev A 45, 3403–3411 (1992)

    Article  Google Scholar 

  14. Kinsner, W. Characterizing chaos through Lyapunov metrics. Second IEEE International Conference on Cognitive Informatics (ICCI'03), 189–201 (2003)

    Google Scholar 

  15. Lehnertz, K., Andrzejak, R.G., Arnhold, J., Kreuz, T., Mormann, F., Rieke, C., Widman, G., Elger, C. Nonlinear EEG analysis in epilepsy: Its possible use for interictal focus localization, seizure antipilation, and prevention. J Clin Neurophysiol 18, 209–222 (2001)

    Article  Google Scholar 

  16. Nair, S. Brain Dynamics and Control with Applications in Epilepsy. Dissertation at the University of Florida, Gainesville, FL (2006)

    Google Scholar 

  17. Packard, N.H., Crutchfield, J.P., Farmer, J.D., Shaw, R.S. Geometry from a time series. Phys Rev Lett 45, 712–715 (1980)

    Article  Google Scholar 

  18. Pardalos, P.M., Boginski, V., Vazakopoulos, A. (eds.) Data Mining in Biomedicine. Springer, New York (2007)

    Google Scholar 

  19. Pardalos, P.M., Chaovalitwongse, W., Iasemidis, L.D., Sackellares, J.C., Shiau, D-S., Carney, P.R., Prokopyev, O.A., Yatsenko, V.A. Seizure warning algorithm based on optimization and nonlinear dynamics. Math Program 101, 365–385 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pardalos, P.M., Principe, J. (eds.) Biocomputing. Kluwer Academic Publishers, Dordrecht (2002)

    Google Scholar 

  21. Pardalos, P.M., Sackellares, C., Carney, P., Iasemidis, L. (eds.) Quantitative Neuroscience. Kluwer Academic Publishers, Dordrecht (2004)

    Google Scholar 

  22. Pardalos, P.M., Sackellares, J.C., Iasemidis, L.D., Yatsenko, V.A., Yang, M., Shiau, D.-S., Chaovalitwongse,W. Statistical information approaches to modelling and detection of the epileptic human brain. Comput Stat Data Anal 43(1), 79–108 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pardalos, P.M., Sackellares, J.C., Yatsenko, V.A., Butenko, S.I. Nonlinear dynamical systems and adaptive filters in biomedicine. Ann Oper Res 119, 119–142 (2003)

    Article  MATH  Google Scholar 

  24. Pardalos, P.M., Yatsenko, V.A. Optimization approach to the estimation and control of Lyapunov exponents. J Optim Theory Appl 128, 29–48 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pardalos, P.M., Yatsenko, V.A., Sackellares, J.C., Shiau, D.-S., Chaovalitwongse, W., Iasemidis, L. Analysis of EEG data using optimization, statistics, and dynamical systems techniques. Comput Stat Data Anal 44, 391–408 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ramasubramanian, K., Sriram, M.S. A comparative study of computation of Lyapunov spectra with different algorithms. Physica D 139, 72–86 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rosenstein, M.T., Collins, J.J., De Luca, C.J. A practical method for calculating largest Lyapunov exponents from small data sets. Physica D 65, 117–134 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sano, M., Sawada, Y. Measurement of the Lyapunov spectrum from a chaotic time series. Phys Rev Lett 55, 1082–1085 (1985)

    Article  MathSciNet  Google Scholar 

  29. Serfaty DeMarkus, A. Detection of the onset of numerical chaotic instabilities by Lyapunov exponents. Discrete Dyn Nat Soc, 6, 121–128 (2001)

    Article  Google Scholar 

  30. Shimada, I., Nagashima, T. A Numerical approach to ergodic problem of dissipative dynamical systems. Progr Theor Phys, 61, 1605–1616 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  31. Strogatz, S.H. Nonlinear Dynamics and Chaos. Perseus Books Publishing, Cambridge, MA (1994)

    Google Scholar 

  32. Takens, F. Detecting strange attractors in turbulence. In: Rand, D.A., Young, L.-S. (eds.) Dynamical Systems and Turbulence, Lecture Notes in Mathematics, Vol. 898, pp. 366–381. Springer-Verlag, New York (1981)

    Chapter  Google Scholar 

  33. Wiesel,W.E. Modal feedback control on chaotic trajectories. Phys Rev E 49, 1990–1996 (1994)

    Article  MathSciNet  Google Scholar 

  34. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A. Determining Lyapunov exponents from a time series. Physica D 16, 285–317 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zeni, A.R., Gallas, J.A.C. Lyapunov exponents for a Duffing oscillator. Physica D 89, 71–82 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, Center for Applied Optimization, University of Florida, Gainesville, FL, USA

    Panos M. Pardalos, Vitaliy A. Yatsenko, Altannar Chinchuluun & Petros Xanthopoulos

  2. Department of Optimization, Kungliga Tekniska Högskolan, Stockholm, Sweden

    Alexandre Messo

Authors
  1. Panos M. Pardalos
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  2. Vitaliy A. Yatsenko
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  3. Alexandre Messo
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  4. Altannar Chinchuluun
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  5. Petros Xanthopoulos
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Corresponding authors

Correspondence to Panos M. Pardalos , Vitaliy A. Yatsenko , Alexandre Messo , Altannar Chinchuluun or Petros Xanthopoulos .

Editor information

Editors and Affiliations

  1. Department of Industrial and Systems, Engineering, Rutgers State University of New Jersey, Frelinghuysen Rd. 96, Piscataway, 08854, New Jersey, USA

    Wanpracha Chaovalitwongse

  2. Dept. Industrial & Systems, Engineering, University of Florida, Weil Hall 303, Gainesville, 32611-6595, Florida, USA

    Panos M. Pardalos

  3. , Dept. of Industrial & Systems Eng., University of Florida, Weil Hall 303, Gainesville, 32611, Florida, USA

    Petros Xanthopoulos

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Pardalos, P.M., Yatsenko, V.A., Messo, A., Chinchuluun, A., Xanthopoulos, P. (2010). An Optimization Approach for Finding a Spectrum of Lyapunov Exponents. In: Chaovalitwongse, W., Pardalos, P., Xanthopoulos, P. (eds) Computational Neuroscience. Springer Optimization and Its Applications(), vol 38. Springer, New York, NY. https://doi.org/10.1007/978-0-387-88630-5_16

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