International Journal of Dynamics and Control

, Volume 7, Issue 4, pp 1157–1172 | Cite as

On maximizing the positive Lyapunov exponent of chaotic oscillators applying DE and PSO

  • Alejandro Silva-Juárez
  • Carlos Javier Morales-Pérez
  • Luis Gerardo de la Fraga
  • Esteban Tlelo-CuautleEmail author
  • José de Jesús Rangel-Magdaleno


Lyapunov exponents are related to the exponentially fast divergence or convergence of nearby orbits in phase space, and they can be used to evaluate the Kaplan–Yorke dimension. In this manner, since the existence of a positive Lyapunov exponent (LE+) is taken as an indication that chaotic behavior exists, and due to the huge search spaces of the design variables of chaotic oscillators, we show the application of differential evolution (DE) and particle swarm optimization (PSO) algorithms to maximize LE+. Four chaotic oscillators are optimized herein, for which we detail the evaluation of their equilibrium points and their eigenvalues that are used to estimate the step-size h to perform appropriate numerical simulation. Both DE and PSO are calibrated to perform different number of generations with three different sizes of individuals in the populations, and with search spaces around the values already published for the four chaotic oscillators. As a result, we show that both DE and PSO algorithms provide higher values of LE+ and Kaplan–Yorke dimension compared to the ones already published in the literature.


Optimization Chaos Lyapunov exponent Kaplan–Yorke dimension Differential evolution Particle swarm optimization Numerical simulation Step-size 


Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Devaney R (2008) An introduction to chaotic dynamical systems. Westview press, BoulderGoogle Scholar
  2. 2.
    Wolf A, Swift JB, Swinney HL, Vastano JA (1985) Determining Lyapunov exponents from a time series. Phys D: Nonlinear Phenom 16(3):285–317MathSciNetCrossRefGoogle Scholar
  3. 3.
    Yang CJ, Zhu WD, Ren GX (2013) Approximate and efficient calculation of dominant lyapunov exponents of high-dimensional nonlinear dynamic systems. Commun Nonlinear Sci Numer Simul 18(12):3271–3277MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dieci L (2002) Jacobian free computation of Lyapunov exponents. J Dyn Differ Equ 14(3):697–717MathSciNetCrossRefGoogle Scholar
  5. 5.
    Rugonyi S, Bathe K-J (2003) An evaluation of the Lyapunov characteristic exponent of chaotic continuous systems. Int J Numer Methods Eng 56(1):145–163MathSciNetCrossRefGoogle Scholar
  6. 6.
    Wilkinson A (2017) What are lyapunov exponents, and why are they interesting? Bull Am Math Soc 54(1):79–105MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kapitaniak T (1992) Chaotic oscillators: theory and applications. World Scientific, SingaporeCrossRefGoogle Scholar
  8. 8.
    Clinton SJ, Sprott JC (2003) Chaos and time-series analysis, vol 69. Oxford University Press, New York zbMATHGoogle Scholar
  9. 9.
    Nguyen VH, Kumar S, Song H (2018) A family of fully integrated CMOS chaos generators with strictly 1-D linear-piecewise chaos maps. J Comput Electron 17(3):1343–1355CrossRefGoogle Scholar
  10. 10.
    Sui Y, He Y, Wenxin Y, Li Y (2018) Design and circuit implementation of a five-dimensional hyperchaotic system with linear parameter. Int J Circuit Theory Appl 46(8):1503–1515CrossRefGoogle Scholar
  11. 11.
    Chen L, Tang S, Li Q, Zhong S (2018) A new 4D hyperchaotic system with high complexity. Math Comput Simul 146:44–56MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hua Z, Yi S, Zhou Y, Li C, Yue W (2018) Designing hyperchaotic cat maps with any desired number of positive Lyapunov exponents. IEEE Trans Cybern 48(2):463–473CrossRefGoogle Scholar
  13. 13.
    Peixoto MLC, Nepomuceno EG, Martins SAM, Lacerda MJ (2018) Computation of the largest positive Lyapunov exponent using rounding mode and recursive least square algorithm. Chaos Solitons Fractals 112:36–43MathSciNetCrossRefGoogle Scholar
  14. 14.
    Shao H, Shi Y, Zhu H (2018) Lyapunov exponents, sensitivity, and stability for non-autonomous discrete systems. Int J Bifurc Chaos 28(07):1850088MathSciNetCrossRefGoogle Scholar
  15. 15.
    Barreira L, Valls C (2018) Transformations preserving the Lyapunov exponents. Commun Contemp Math 20(04):1750027MathSciNetCrossRefGoogle Scholar
  16. 16.
    Nguyen KC, Nhung T, Anh Hoa TT, Liem NC (2018) Lyapunov exponents for dynamic equations on time scales. Dyn Syst Appl 27(2):367–386Google Scholar
  17. 17.
    Pano-Azucena AD, Tlelo-Cuautle E, Rodriguez-Gomez G, de la Fraga LG (2018) FPGA-based implementation of chaotic oscillators by applying the numerical method based on trigonometric polynomials. AIP Adv 8(7):075217CrossRefGoogle Scholar
  18. 18.
    Tlelo-Cuautle E, de la Fraga LG, Pham V-T, Volos C, Jafari S, de Jesus Quintas-Valles A (2017) Dynamics, FPGA realization and application of a chaotic system with an infinite number of equilibrium points. Nonlinear Dyn 89(2):1129–1139CrossRefGoogle Scholar
  19. 19.
    Lu J, Chen G, Yu X, Leung H (2004) Design and analysis of multiscroll chaotic attractors from saturated function series. IEEE Trans Circuits Syst I: Regul Pap 51(12):2476–2490MathSciNetCrossRefGoogle Scholar
  20. 20.
    Rajagopal K, Akgul A, Moroz IM, Wei Z, Jafari S, Hussain I (2019) A simple chaotic system with topologically different attractors. IEEE Access 7:89936–89947CrossRefGoogle Scholar
  21. 21.
    Volos CK, Jafari S, Kengne J, Munoz-Pacheco JM, Rajagopal K (2019) Nonlinear dynamics and entropy of complex systems with hidden and self-excited attractors. Entropy 21(4):370MathSciNetCrossRefGoogle Scholar
  22. 22.
    Dellnitz M, Hohmann A (1996) The computation of unstable manifolds using subdivision and continuation. In: Nonlinear dynamical systems and chaos. Springer, Berlin, pp 449–459CrossRefGoogle Scholar
  23. 23.
    Dellnitz M, Hohmann A (1997) A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer Math 75(3):293–317MathSciNetCrossRefGoogle Scholar
  24. 24.
    Dellnitz M, Junge O (1997) Almost invariant sets in Chua’s circuit. Int J Bifurc Chaos 7(11):2475–2485MathSciNetCrossRefGoogle Scholar
  25. 25.
    Dellnitz M, Junge O (1999) On the approximation of complicated dynamical behavior. SIAM J Numer Anal 36(2):491–515MathSciNetCrossRefGoogle Scholar
  26. 26.
    Xiong F-R, Qin Z-C, Ding Q, Hernández C, Fernandez J, Schütze O, Sun J-Q (2015) Parallel cell mapping method for global analysis of high-dimensional nonlinear dynamical systems. J Appl Mech 82(11):111010CrossRefGoogle Scholar
  27. 27.
    Sun J-Q, Xiong F-R, Schutze O (2019) Cell mapping methods: algorithmic approaches and applications. Springer, Berlin CrossRefGoogle Scholar
  28. 28.
    Sprott JC, Xiong A (2015) Classifying and quantifying basins of attraction. Chaos: Interdiscip J Nonlinear Sci 25(8):083101MathSciNetCrossRefGoogle Scholar
  29. 29.
    Pham V-T, Vaidyanathan S, Volos C, Kapitaniak T (2018) Nonlinear dynamical systems with self-excited and hidden attractors, vol 133. Springer, BerlinCrossRefGoogle Scholar
  30. 30.
    Rössler OE (1976) An equation for continuous chaos. Phys Lett A 57(5):397–398CrossRefGoogle Scholar
  31. 31.
    Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20(2):130–141CrossRefGoogle Scholar
  32. 32.
    Li S-Y, Huang S-C, Yang C-H, Ge Z-M (2012) Generating tri-chaos attractors with three positive Lyapunov exponents in new four order system via linear coupling. Nonlinear Dyn 69(3):805–816MathSciNetCrossRefGoogle Scholar
  33. 33.
    Sun Y, Wu CQ (2012) A radial-basis-function network-based method of estimating Lyapunov exponents from a scalar time series for analyzing nonlinear systems stability. Nonlinear Dyn 70(2):1689–1708MathSciNetCrossRefGoogle Scholar
  34. 34.
    Li C, Sprott JC (2013) Amplitude control approach for chaotic signals. Nonlinear Dyn 73(3):1335–1341MathSciNetCrossRefGoogle Scholar
  35. 35.
    Storn R, Price K (1997) Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11(4):341–359MathSciNetCrossRefGoogle Scholar
  36. 36.
    Kennedy J (2010) Particle swarm optimization. Springer, Boston, pp 760–766Google Scholar
  37. 37.
    Parker TS, Chua L (2012) Practical numerical algorithms for chaotic systems. Springer, BerlinzbMATHGoogle Scholar
  38. 38.
    Cardano G, Witmer TR (1968) Ars magna or the rules of algebra. Dover Books on Advanced Mathematics, DoverGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.INAOETonantzintlaMexico
  2. 2.Computer Science DepartmentCinvestavMexico CityMexico

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