Calculation of the Largest Lyapunov Exponent

The calculation of the largest Lyapunov exponent is a crucial method for analyzing time series data. The maximum Lyapunov exponent is computed using Wolf's method, which requires careful consideration of the following parameters: - Embedding dimension (m): Determines the dimensionality for phase space reconstruction from the time series. In code implementation, this typically involves using delay coordinates to create multidimensional vectors. - Time delay (tau): Specifies the interval between consecutive points in the reconstructed phase space. Optimal tau selection can be achieved using methods like mutual information or autocorrelation function analysis. - Time series (data): The input data sequence for which the Lyapunov exponent is calculated. The data should be preprocessed to remove trends and normalize if necessary. - Time series length (N): The total number of data points in the time series. Sufficient data length is critical for accurate exponent calculation. - Average period (P): Defines the minimum distance requirement when searching for evolution phase points. When the current phase point is I, the algorithm must search for evolution points J that satisfy |I-J|>P to avoid tracking adjacent orbits. A key implementation detail involves maintaining proper separation between current and evolution points to ensure meaningful divergence measurements. The algorithm typically tracks the exponential divergence of nearby trajectories in phase space over time, requiring careful neighbor searching and distance ratio calculations at each iteration step.