Mathematical Software - Chaotic Systems - Lyapunov Exponent

Lyapunov Exponent

The Largest Positive Lyapunov Exponent - Definition

The algorithm of Wolf et al. (Physica D 16, 285-317) allows to compute the largest positive Lyapunov exponent from a single time series of K time points. This method is based on examining a distance divergence between a chosen trajectory called furher the central trajectory and another one close enough to the given trajectory. Inspection of trajectories behavior is done along the central trajectory each M time points. M value must be set by an user. After a total time t_N-t₀ the magnitude of trajectories divergence is given by the expression

	N - 1
λ =	∑	ln(D_(i+1)/D^old_i) /(t_N - t₀)
	i = 0

where D is a new distance between two trajectories chosen as the neighbouring ones whereas D^old is the previous value of trajectories discrepancy i. e. the value before time shift. Because chaotic trajectories diverge very quickly after each time shift a new close neighbouring trajectory is found. The new one replaces the old trajectory. The largest positive Lyapunov exponent allows to estimate the predictability time

T = (1/λ)(1/d)lnK

where d is the attractor dimension and K denotes the number of points on the attractor.

Chaotic Systems - References

^{^} G. L. Baker, J. P. Gollub, Chaotic dynamics: an introduction, Cambridge University Press, 1996
^{^} Vadim S. Anishchenko et al., Nonlinear Dynamics of Chaotic and Stochastic Systems, Springer-Verlag, 2007
^{^} Boris P. Bezruchko and Dmitry A. Smirnov, Extracting Knowledge From Time Series, Springer-Verlag, 2010

Machine Learning - OptFinderML

Package for machine learning - OptFinderML.