Lyapunov exponent
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The Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a measure that characterizes rate of separation of infinitesimally close trajectories. Quantitatively, in chaotic system 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 embedding phase space. It is common to just refer to the largest one, because it determines the predictability of a dynamical system.
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Definition
Mentioned notion leads to definition of Lyapunov exponent
- <math>\lambda_i = \lim_{t \to \infty} \lim_{\left\| \delta Z_0 \right\| \to 0}
\frac{1}{t} \ln \left( \frac{\left\| \delta\mathbf{Z}(t) \right\|}{\left\| \delta \mathbf{Z}_0 \right\|} \right), <math>
where <math>\delta \mathbf{Z}_0<math> is the initial distance in phase-space between a reference solution and a perturbation, and <math>\delta \mathbf{Z}(t)<math> is the distance at time t. For each choice of direction for the initial perturbation a different Lyapunov exponent is given.
In other commonly used definition, Lyapunov exponents λi are calculated as
- <math>\lambda_i = \lim_{t \to \infty} \frac{1}{t} \ln \left( \frac{d L_i(t)}{d r} \right),<math>
which can be thought of as following the motion of an infinitesimally small sphere, with an initial radius dr, that starts from the point for which the exponent should be calculated. On its trajectory, it will get "squished" unevenly, so that it becomes an ellipsoid with time-dependent radii dLi(t) in each principal direction.
Basic properties
If the system is conservative (i.e. there is no dissipation), volume of an element of the phase space will stay the same along the trajectory. Thus the sum of all Lyapunov exponents is zero. If the system is dissipative, the sum of Lyapunov exponents is negative.
It can be easily seen that if the system is a flow, one exponent is always zero - if the initial separation <math> \mathbf{Z}_0 <math> is in the direction of unperturbed solution, the trajectories will never depart.
Significance of Lyapunov spectrum
Lyapunov spectrum can be used as a tool in analysis of a system.
Consider example of 3 dimensional dissipative system, e.g. well known Lorenz system. According to general properties of Lyapunov exponents, systems may be classified in three cases
- <math> 0 > \lambda_1 > \lambda_2 <math> Both remaining Lyapunov exponents are negative. System is integrable.
- <math>0 = \lambda_1 > \lambda_2 <math> One remaining Lyapunov exponents are negative. System is integrable.
- <math> \lambda_1 > 0 > \lambda_2 <math> The largest exponent is positive, the system is [[|chaos|chaotic]].
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, Lyapunov time will be finite, whereas for regular orbits it will be infinite.
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 calculate the largest Lyapunov exponents and recover the others by means of the Gram-Schmidt process.
See also
Reference
Cvitanović P., Artuso R., Mainieri R. , Tanner G. and Vattay G.Chaos: Classical and Quantum (http://www.chaosbook.org/) Niels Bohr Institute, Copenhagen 2005 - textbook about chaos available under Free Documentation License
Sprott J. C. Chaos and Time-Series Analysis (http://sprott.physics.wisc.edu/chaostsa/) Oxford University Press, 2003 - see also online supplement Numerical Calculation of Largest Lyapunov Exponent (http://sprott.physics.wisc.edu/chaos/lyapexp.htm)de:Ljapunow-Exponent