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Characterization and Optimization of Fractional-Order Chaotic Systems

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Analog/Digital Implementation of Fractional Order Chaotic Circuits and Applications

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Abstract

Guaranteeing chaotic behavior in integer and fractional-order chaotic systems for long-time simulation is a challenge. This chapter details the requirements for a correct time simulation of an integer and fractional-order chaotic system, in order to evaluate the Lyapunov exponents and Kaplan–Yorke dimension. Using TISEAN or Wolf’s method one can compute both Lyapunov exponents and Kaplan–Yorke dimension within an optimization loop that can be performed by applying metaheuristics. Differential evolution and particle swarm optimization algorithms are applied herein to optimize the positive Lyapunov exponent and Kaplan–Yorke dimension of integer and fractional-order chaotic oscillators.

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References

  1. I. Grigorenko, E. Grigorenko, Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett. 91(3), 034101 (2003)

    Google Scholar 

  2. J. Lü, G. Chen, A new chaotic attractor coined. Int. J. Bifurcation Chaos 12(3), 659–661 (2002)

    Article  MathSciNet  Google Scholar 

  3. M.F. Tolba, A.M. AbdelAty, N.S. Soliman, L.A. Said, A.H. Madian, A.T. Azar, A.G. Radwan, FPGA implementation of two fractional order chaotic systems. AEU-Int. J. Electron. Commun. 78, 162–172 (2017)

    Article  Google Scholar 

  4. G. Cardano, T.R. Witmer, Ars Magna or the Rules of Algebra. Dover Books on Advanced Mathematics (Dover, New York, 1968)

    Google Scholar 

  5. R. Garrappa, Short tutorial: solving fractional differential equations by Matlab codes. Department of Mathematics, University of Bari (2014)

    Google Scholar 

  6. M.-F. Danca, N. Kuznetsov, Matlab code for Lyapunov exponents of fractional-order systems. Int. J. Bifurcation Chaos 28(5), 1850067 (2018)

    Article  MathSciNet  Google Scholar 

  7. A.D. Pano-Azucena, E. Tlelo-Cuautle, G. Rodriguez-Gomez, L.G. De la Fraga, FPGA-based implementation of chaotic oscillators by applying the numerical method based on trigonometric polynomials. AIP Adv. 8(7), 075217 (2018)

    Article  Google Scholar 

  8. A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining Lyapunov exponents from a time series. Phys. D Nonlinear Phenomena 16(3), 285–317 (1985)

    Article  MathSciNet  Google Scholar 

  9. A. Bespalov, N. Polyakhov, Determination of the largest Lyapunov exponents based on time series. World Appl. Sci. J. 26(2), 157–164 (2013)

    Google Scholar 

  10. K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type (Springer, New York, 2010)

    Book  Google Scholar 

  11. K. Diethelm, N.J. Ford, A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynam. 29(1–4), 3–22 (2002)

    Article  MathSciNet  Google Scholar 

  12. R. Garrappa, Predictor-corrector PECE method for fractional differential equations. MATLAB Central File Exchange [File ID: 32918] (2011)

    Google Scholar 

  13. M.-F. Danca, N. Kuznetsov, Matlab code for Lyapunov exponents of fractional-order systems. Int. J. Bifurcation Chaos 28(5), 1850067 (2018)

    Article  MathSciNet  Google Scholar 

  14. H. Kantz, A robust method to estimate the maximal Lyapunov exponent of a time series. Phys. Lett. A 185(1), 77–87 (1994)

    Article  Google Scholar 

  15. M.T. Rosenstein, J.J. Collins, C.J. De Luca, A practical method for calculating largest Lyapunov exponents from small data sets. Phys. D Nonlinear Phenomena 65(1–2), 117–134 (1993)

    Article  MathSciNet  Google Scholar 

  16. R. Hegger, H. Kantz, T. Schreiber, Practical implementation of nonlinear time series methods: the TISEAN package. Chaos: An Interdisciplinary J. Nonlinear Sci. 9(2), 413–435 (1999)

    Article  Google Scholar 

  17. J. Petržela, Optimal piecewise-linear approximation of the quadratic chaotic dynamics. Radioengineering 21(1), 20–28 (2012)

    Google Scholar 

  18. S.-Y. Li, S.-C. Huang, C.-H. Yang, Z.-M. Ge, Generating tri-chaos attractors with three positive Lyapunov exponents in new four order system via linear coupling. Nonlinear Dynam. 69(3), 805–816 (2012)

    Article  MathSciNet  Google Scholar 

  19. Y. Sun, C.Q. Wu, A radial-basis-function network-based method of estimating Lyapunov exponents from a scalar time series for analyzing nonlinear systems stability. Nonlinear Dynam. 70(2), 1689–1708 (2012)

    Article  MathSciNet  Google Scholar 

  20. C. Li, J.C. Sprott, Amplitude control approach for chaotic signals. Nonlinear Dynam. 73(3), 1335–1341 (2013)

    Article  MathSciNet  Google Scholar 

  21. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, Hoboken, 1993)

    MATH  Google Scholar 

  22. K. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, vol. 111 (Elsevier, Amsterdam, 1974)

    MATH  Google Scholar 

  23. M. Caputo, Linear models of dissipation whose q is almost frequency independent-II. Geophys. J. Int. 13(5), 529–539 (1967)

    Article  Google Scholar 

  24. L. Dorcak, J. Prokop, I. Kostial, Investigation of the properties of fractional-order dynamical systems, in Proceedings of 11th International Conference on Process Control (1994), pp. 19–20

    Google Scholar 

  25. D. Cafagna, G. Grassi, On the simplest fractional-order memristor-based chaotic system. Nonlinear Dynam. 70(2), 1185–1197 (2012)

    Article  MathSciNet  Google Scholar 

  26. R. Storn, K. Price, Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11(4), 341–359 (1997)

    Article  MathSciNet  Google Scholar 

  27. R. Kennedy, J. Eberhart, Particle swarm optimization, in Proceedings of IEEE International Conference on Neural Networks IV, vol. 1000 (1995)

    Google Scholar 

  28. C.J. Yang, W.D. Zhu, G.X. Ren, Approximate and efficient calculation of dominant Lyapunov exponents of high-dimensional nonlinear dynamic systems. Commun. Nonlinear Sci. Numer. Simul. 18(12), 3271–3277 (2013)

    Article  MathSciNet  Google Scholar 

  29. L. Dieci, Jacobian free computation of Lyapunov exponents. J. Dyn. Diff. Equ. 14(3), 697–717 (2002)

    Article  MathSciNet  Google Scholar 

  30. S. Rugonyi, K.-J. Bathe, An evaluation of the Lyapunov characteristic exponent of chaotic continuous systems. Int. J. Numer. Methods Eng. 56(1), 145–163 (2003)

    Article  MathSciNet  Google Scholar 

  31. J.L. Kaplan, J.A. Yorke, Chaotic behavior of multidimensional difference equations, in Functional Differential Equations and Approximation of Fixed Points (Springer, Berlin, 1979), pp. 204–227

    Book  Google Scholar 

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Authors and Affiliations

  1. INAOE, Tonantzintla, Puebla, Mexico

    Esteban Tlelo-Cuautle, Ana Dalia Pano-Azucena, Omar Guillén-Fernández & Alejandro Silva-Juárez

Authors
  1. Esteban Tlelo-Cuautle
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  2. Ana Dalia Pano-Azucena
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  3. Omar Guillén-Fernández
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  4. Alejandro Silva-Juárez
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Tlelo-Cuautle, E., Dalia Pano-Azucena, A., Guillén-Fernández, O., Silva-Juárez, A. (2020). Characterization and Optimization of Fractional-Order Chaotic Systems. In: Analog/Digital Implementation of Fractional Order Chaotic Circuits and Applications. Springer, Cham. https://doi.org/10.1007/978-3-030-31250-3_3

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  • DOI: https://doi.org/10.1007/978-3-030-31250-3_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-31249-7

  • Online ISBN: 978-3-030-31250-3

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