Lyapunov exponent

From Free net encyclopedia

The Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation <math>\delta \mathbf{Z}_0</math> diverge

<math> | \delta\mathbf{Z}(t) | \approx e^{\lambda t} | \delta \mathbf{Z}_0 |</math>

The rate of separation can be different for different orientations of initial separation vector. Thus, there is whole spectrum of Lyapunov exponents—the number of them is equal to the number of dimensions of the phase space. It is common to just refer to the largest one, because it determines the predictability of a dynamical system.

Contents

[edit]

Definition

For a dynamical system with evolution equation <math>f^t</math> in a n–dimensional phase space, the spectrum of Lyapunov exponents

<math> \{ \lambda_1, \lambda_2, \cdots , \lambda_n \} \,, </math>

in general, depends on the starting point <math>x_0</math>. The Lyapunov exponents describe the behavior of vectors in the tangent space of the phase space and are defined from the Jacobian matrix

<math> J^t(x_0) = \left. \frac{ d f^t(x) }{dx} \right|_{x_0}. </math>

The <math>J^t</math> matrix describes how a small change at the point <math>x_0</math> propagates to the final point <math>f^t(x_0)</math>. The limit

<math> \lim_{t \rightarrow \infty} (J^t \cdot \mathrm{Transpose}(J^t) )^{1/2t} </math>

defines a matrix <math>L(x_0)</math> (the conditions for the existence of the limit are given by the Oseldec theorem). If <math> \Lambda_i(x_0) </math> are the eigenvalues of <math>L(x_0)</math>, then the Lyapunov exponents <math>\lambda_i</math> are defined by

<math> \lambda_i(x_0) = \log \Lambda_i(x_0).\,</math>

The set of Lyapunov exponents will be the same for almost all stating points of an ergodic component of the dynamical system.

[edit]

Basic properties

If the system is conservative (i.e. there is no dissipation), a volume element of the phase space will stay the same along a trajectory. Thus the sum of all Lyapunov exponents must be zero. If the system is dissipative, the sum of Lyapunov exponents is negative.

If the system is a flow, one exponent is always zero—the Lyapunov exponent corresponding to the eigenvalue of <math>L</math> with an eigenvector in the direction of the flow.

[edit]

Significance of the Lyapunov spectrum

The Lyapunov spectrum can be used to estimate the rate of entropy production of the dynamical system.

The inverse of the largest Lyapunov exponent is sometimes referred in literature as Lyapunov time, and defines the characteristic e-folding time. For chaotic orbits, the Lyapunov time will be finite, whereas for regular orbits it will be infinite.

[edit]

Numerical calculation

Generally the calculation of Lyapunov exponents, as defined above, cannot be carried out analytically, and in most cases one must resort to numerical techniques. The commonly used numerical procedures estimates the <math>L</math> matrix based on averaging several finite time approximations of the limit defining <math>L</math>.

For the calculation of Lyapunov exponents from limited experimental data, various methods have been proposed [1].

[edit]

Local Lyapunov exponent

Whereas the (global) Lyapunov exponent gives a measure for the total predictability of a system, it is sometimes interesting to estimate the local predictability around a point x0 in phase space. This may be done through the eigenvalues of the Jacobian matrix J 0(x0). These eigenvalues are also called local Lyapunov exponents. The eigenvectors of the Jacobian matrix point in the direction of the stable and unstable manifolds.

[edit]

See also

[edit]

References

  • {{cite journal 
| author = Freddy Christiansen and Hans Henrik Rugh
| year = 1997
| title = Computing Lyapunov spectra with continuous Gram-Schmidt orthonormalization
| journal = Nonlinearity
| volume = 10
| pages = 1063–1072
| url = http://www.mpipks-dresden.mpg.de/eprint/freddy/9702017/9702017.ps
}}
  • {{cite journal 
| author = Govindan Rangarajan, Salman Habib, and Robert D. Ryne
| year = 1998
| title = Lyapunov Exponents without Rescaling and Reorthogonalization
| journal = Physical Review Letters
| volume = 80
| pages = 3747–3750
| url = http://arxiv.org/pdf/chao-dyn/9803017
}}
  • {{cite journal 
| author = X. Zeng, R. Eykholt, and R. A. Pielke
| year = 1991
| title = Estimating the Lyapunov-exponent spectrum from short time series of low precision
| journal = Physical Review Letters
| volume = 66
| pages = 3229
| url = http://link.aps.org/abstract/PRL/v66/p3229
}}de:Ljapunow-Exponent

it:Esponente di Lyapunov pl:Wykładnik Lapunowa

Retrieved from "http://www.netipedia.com/index.php/Lyapunov_exponent"