On the Numerical Value of Finite-Time Pseudo-Lyapunov Exponents

  • Franziska Schmidt
  • Claude-Henri Lamarque
Conference paper
Lyapunov exponents measure the sensitivity of a dynamical system to initial conditions [1,2]. In fact an infinitesimal difference of initial conditions may lead to totally different paths. This is the case if the computed Lyapunov exponent is strictly positive. The system is then theoretically chaotic in infinite time, but practically this may occur at finite time considered as “asymptotic” for applications. Mathematically, the numerical value of the Lyapunov exponents is given by the formula:
$$\lambda = \mathop {\lim \;\;\lim }\limits_{t \to + \infty \left\| {\delta x_0 } \right\| \to 0t} \frac{1}{t}in\left({\frac{{\left\| {\delta x\left({t,x_0 } \right)} \right\|}}{{\left\| {\delta x_0 } \right\|}}} \right)$$
where x 0 is the initial condition (it may be a vector of initial conditions in ℝn), t is the time, ∥̤∥ is a norm for ℝ n , ∂x 0 is an infinitesimal divergence in initial conditions (in ℝ n ) and .x (∈ ℝ n ) is the path followed by the system starting at x = x 0 at time t =0.

The maximum of this spectrum is often the only one that is computed to detect chaos.


Lyapunov Exponent Jacobian Matrix Lorenz System Linear Oscillator Lyapunov Spectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    A. M. Lyapunov. The general problem of the stability of motion. .International Journal of Control., 55(3):531–773, 1992.CrossRefMathSciNetGoogle Scholar
  2. 2.
    V. I. Oseledec. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Transactions of the Moscow Mathematical Society, 19:197–231, 1968.MathSciNetGoogle Scholar
  3. 3.
    A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano. Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena, 16(3):285–317, 1985.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    F. E. Udwadia and H. F. von Bremen. An efficient and stable approach for computation of Lyapunov characteristic exponents of continuous dynamical systems. Applied Mathematics and Computation, 121(2–3):219–259, 2001.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    K. Ramasubramanian and M. S. Sriram. A comparative study of computation of Lyapunov spectra with different algorithms. Physica D: Nonlinear Phenomena, 139(1–2):72–86, 2000.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    G. Tancredi, A. Sànchez, and F. Roig. A comparison between methods to compute Lyapunov exponents. Astronomical Journal, 121(2):1171–1179, 2001.CrossRefGoogle Scholar
  7. 7.
    R. Ding and J. Li. Nonlinear finite-time Lyapunov exponent and predictability. Physics Letters A, 364(5):396–400, 2007.CrossRefMathSciNetGoogle Scholar
  8. 8.
    E. Aurell, G. Boffetta, A. Crisanti, G. Paladin, and A. Vulpiani. Growth of non-infinitesimal perturbations in turbulence. Physical Review Letters, 77(7):1262–1265, 1996.CrossRefGoogle Scholar
  9. 9.
    J. M. Nese. Quantifying local predictability in phase space. Physica D: Nonlinear Phenomena, 35(1–2):237–250, 1989.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    C. Nicolis, S. Vannitsem, and J. -F. Royer. Short-range predictability of the atmosphere: mechanisms for superexponential error growth. Quarterly Journal — Royal Meteorological Society, 121(523):705–722, 1995.CrossRefGoogle Scholar
  11. 11.
    N. Hinrichs, M. Oestreich, and K. Popp. Dynamics of oscillators with impact and friction. Chaos, Solitons & Fractals Nonlinearities in Mechanical Engineering, 8(4):535–558, 1997.MATHCrossRefGoogle Scholar
  12. 12.
    L. Jin, Q. -S. Lu, and E. H. Twizell. A method for calculating the spectrum of Lyapunov exponents by local maps in non-smooth impact-vibrating systems. Journal of Sound and Vibration, 298(4–5):1019–1033, 2006.CrossRefMathSciNetGoogle Scholar
  13. 13.
    P. C. Muller. Calculation of Lyapunov exponents for dynamic systems with discontinuities. Chaos, Solitons &Fractals Some Nonlinear Oscillations Problems in Engineering Sciences, 5(9):1671–1681, 1995.Google Scholar
  14. 14.
    A. Stefański and T. Kapitaniak. Estimation of the dominant Lyapunov exponent of non-smooth systems on the basis of maps synchronization. Chaos, Solitons & Fractals, 15(2):233– 244, 2003.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Q. Wu and N. Sepehri. On Lyapunov's stability analysis of non-smooth systems with applications to control engineering. International Journal of Non-linear Mechanics, 36(7):1153– 1161, 2001.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    F. Schmidt and C. -H. Lamarque. Computation of the solutions of the Fokker-Planck equation for one and two dof systems. Communications in Nonlinear Science and Numerical Simulation, 74(2008):529–542.Google Scholar
  17. 17.
    F.Schmidt and C.-H.Lamarque. Un indicateur pour optimiser les calculs trajectographiques. Bulletin de Liaison des Ponts et Chaussées, BLPC No. 263–264, znillet — Qoût — septembre 2006.Google Scholar
  18. 18.
    A. P. Ivanov. The dynamics of systems near to grazing collision. Journal of Applied Mathematics and Mechanics, 58(3):437–444, 1994.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    O. Janin and C. H. Lamarque. Stability of singular periodic motions in a vibro-impact oscillator. Nonlinear Dynamics, 28(3–4):231–241, 2002.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    U. Dressler and J. D. Farmer. Generalized Lyapunov exponents corresponding to higher derivatives. Physica D: Nonlinear Phenomena, 59(4):365–377, 1992.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2009

Authors and Affiliations

  1. 1.Université de LyonLyonFrance
  2. 2.URA 1652, Département GénieCivil et Batiment.Ecole Nationale des Travaux Publics de l'Etat, CNRSVaulx-en-VelinFrance

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