Nonlinear Dynamics

, Volume 67, Issue 1, pp 413–424 | Cite as

A method for Lyapunov spectrum estimation using cloned dynamics and its application to the discontinuously-excited FitzHugh–Nagumo model

  • Diogo C. Soriano
  • Filipe I. Fazanaro
  • Ricardo Suyama
  • José Raimundo de Oliveira
  • Romis Attux
  • Marconi K. Madrid
Original Paper


This work presents a new method to calculate the Lyapunov spectrum of dynamical systems based on the time evolution of initially small disturbed copies (“clones”) of the motion equations. In this approach, it is not necessary to construct the tangent space associated with the time evolution of linearized versions of motion equations, being the Lyapunov exponents directly estimated in terms of the rate of convergence or divergence of these disturbed clones with respect to the fiducial trajectory, there being periodic correction via the Gram–Schmidt Reorthonormalization procedure. The proposed method offers the possibility of partial estimation of the Lyapunov spectrum and can also be applied to nonsmooth dynamics, since the linearization procedure is no longer required. The idea is tested for representative continuous- and discrete-time dynamical systems and validated by means of comparison with the classical method to perform this calculation. To illustrate its applicability in the nonsmooth context, the largest Lyapunov exponent of the FitzHugh–Nagumo neuronal model under discontinuous periodic excitation is calculated taking the amplitude of stimulation as control parameter. This analysis reveals some complex behaviours for this simple neuronal model, which motivates relevant discussions about the possible role of chaos in the cognitive process.


Chaos Lyapunov spectrum estimation FitzHugh–Nagumo neuronal model Discontinuous excitation Nonsmooth models 


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  1. 1.
    Abarbanel, H.D.I., Brown, R., Kennel, M.B.: Variation of Lyapunov exponents on a strange attractor. J. Nonlinear Sci. 1(2), 175–199 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Abarbanel, H.D.I.: Analysis of Observed Chaotic Data, Institute for Nonlinear Science, 1st edn. Springer, New York (1996) CrossRefGoogle Scholar
  3. 3.
    Aihara, K., Matsumoto, G., Ikegaya, Y.: Periodic and nonperiodic responses of a periodically forced Hodgkin and Huxley oscillator. J. Theor. Biol. 109(2), 249–269 (1984) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Anishchenko, V.S., Astakhow, V., Neiman, A., Vadivasova, T., Schimansky-Geier, L.: Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments, Springer Series in Synergetics, 2nd edn. Springer, New York (2007) Google Scholar
  5. 5.
    Baptista, M.S., Macau, E.E., Grebogi, C.: Conditions for efficient chaos-based communication. Chaos 13(1), 145–150 (2003) CrossRefGoogle Scholar
  6. 6.
    Bennettin, G., Galgani, L., Strecyn, J.M.: Kolmogorov entropy and numerical experiments. Phys. Rev. A 14(6), 2338–2345 (1976) CrossRefGoogle Scholar
  7. 7.
    Bennettin, G., Galgani, L., Giorgilli, A., Strecyn, J.M.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: a method for computing all of them. Part 2: numerical application. Meccanica 15(2), 21–30 (1980) CrossRefGoogle Scholar
  8. 8.
    Boffetta, G., Cencini, M., Falcioni, M., Vulpiani, A.: Predictability: a way to characterize complexity. Phys. Rep. 356(6), 367–474 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Doi, S., Sato, S.: The global bifurcation structure of the BVP neuronal model driven by periodic pulse trains. Math. Biosci. 125(2), 229–250 (1995) zbMATHCrossRefGoogle Scholar
  10. 10.
    Eckmann, J.-P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57(3), 617–656 (1985) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Eckmann, J.-P., Kamphorst, S.O., Ruelle, D., Ciliberto, S.: Lyapunov exponents from time series. Phys. Rev. A 34(6), 4971–4979 (1986) MathSciNetCrossRefGoogle Scholar
  12. 12.
    FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1(6), 445–466 (1961) CrossRefGoogle Scholar
  13. 13.
    Fujisaka, H., Yamada, T.: Stability theory of synchronized motion in coupled-oscillator systems. Prog. Theor. Phys. 69(1), 32–47 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kaneko, K., Tsuda, I.: Chaotic itinerancy. Chaos 13(3), 926–936 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kinsner, W.: Characterizing chaos through Lyapunov metrics. IEEE Trans. Syst. Man Cybern., Part C, Appl. Rev. 36(2), 141–151 (2006) CrossRefGoogle Scholar
  16. 16.
    Korn, H., Faure, P.: Is there chaos in the brain? II. Experimental evidence and related models. C. R. Biol. 326(9), 787–840 (2003) CrossRefGoogle Scholar
  17. 17.
    Müller, P.C.: Calculation of Lyapunov exponent for dynamic systems with discontinuities. Chaos Solitons Fractals 5(9), 1671–1681 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Oseledec, V.: A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Mosc. Math. Soc. 19, 197–231 (1968) MathSciNetGoogle Scholar
  19. 19.
    Parker, T.S., Chua, L.O.: Practical Numerical Algorithms for Chaotic Systems. Springer, Berlin (1989) zbMATHGoogle Scholar
  20. 20.
    Pomeau, Y., Manneville, P.: Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74(2), 189–197 (1980) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Rabinovich, M.I., Abarbanel, H.D.I.: The role of chaos in neural systems. Neuroscience 87(1), 5–14 (1998) CrossRefGoogle Scholar
  22. 22.
    Ramasubramanian, K., Sriram, M.S.: A comparative study of computation of Lyapunov spectra with different algorithms. Physica D 139(1–2), 72–86 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Shaw, R.: Strange attractors, chaotic behavior and information flow. Z. Naturforsch. A 36, 80–112 (1981) MathSciNetzbMATHGoogle Scholar
  24. 24.
    Shimada, I., Nagashima, T.: A numerical approach to ergodic problem of dissipative dynamical systems. Prog. Theor. Phys. 61(6), 1605–1616 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Skarda, C.A., Freeman, W.J.: How brains make chaos in order to make sense of the world. Behav. Brain Sci. 10, 161–195 (1987) CrossRefGoogle Scholar
  26. 26.
    Stefanski, A.: Estimation of the largest Lyapunov exponent in systems with impacts. Chaos Solitons Fractals 11(15), 2443–2451 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Tanaka, G., Ibarz, B., Sanjuan, M.A.F., Aihara, K.: Synchronization and propagation of bursts in networks of coupled map neurons. Chaos 16(1), 013113-1–013113-10 (2006) MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wang, W.: Bifurcations and chaos of the Bonhoeffer–van der Pol model. J. Phys. A, Math. Gen. 22(13), L627–L632 (1989) CrossRefGoogle Scholar
  29. 29.
    Willians, G.P.: Chaos Theory Tamed. Joseph Henry Press, Atlanta (1997) Google Scholar
  30. 30.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D 16(3), 285–317 (1985) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Diogo C. Soriano
    • 1
  • Filipe I. Fazanaro
    • 2
  • Ricardo Suyama
    • 3
  • José Raimundo de Oliveira
    • 2
  • Romis Attux
    • 1
  • Marconi K. Madrid
    • 4
  1. 1.Laboratory of Signal Processing for Communications (DSPCom), Department of Computer Engineering and Industrial Automation (DCA), School of Electrical and Computer Engineering (FEEC)UNICAMPCampinasBrazil
  2. 2.Modular Robotic Systems Laboratory (LSMR), DCA, FEECUNICAMPCampinasBrazil
  3. 3.Engineering, Modeling and Applied Social Sciences CenterUFABCSanto AndréBrazil
  4. 4.LSMR, Department of Energy Control and Systems (DSCE), FEECUNICAMPCampinasBrazil

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