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 tN-t0 the magnitude of trajectories divergence is given by the expression
|N - 1
|ln(D(i+1)/Doldi) /(tN - t0)
|i = 0
where D is a new distance between two trajectories chosen as the neighbouring ones whereas Dold 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
where d is the attractor dimension and K denotes the number of points on the attractor.
Package for machine learning - OptFinderML.