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}-t_{0} the magnitude of trajectories divergence is given by the expression
N - 1 | ||
λ = | ∑ | ln(D_{(i+1)}/D^{old}_{i}) /(t_{N} - t_{0}) |
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
where d is the attractor dimension and K denotes the number of points on the attractor.
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