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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:
$$\lambda = \mathop {\lim \;\;\lim }\limits_{t \to + \infty \left\| {\delta x_0 } \right\| \to 0t} \frac{1}{t}in\left({\frac{{\left\| {\delta x\left({t,x_0 } \right)} \right\|}}{{\left\| {\delta x_0 } \right\|}}} \right)$$
(1)
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.

Keywords

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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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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