MATLAB Source Code for Computing Maximum Lyapunov Exponent

The maximum Lyapunov exponent is a crucial indicator for characterizing chaotic behavior in nonlinear systems, quantifying the exponential divergence rate of adjacent trajectories in phase space. In MATLAB, this exponent can be computed using numerical methods, typically employing algorithms proposed by Wolf et al. or improved approaches based on small data sets. The implementation often involves matrix operations and iterative calculations that leverage MATLAB's built-in functions for efficient computation.

The computational procedure generally includes: First, preprocessing time series data through operations like noise reduction and phase space reconstruction using embedding techniques. This can be implemented using MATLAB's signal processing toolbox functions such as detrend or custom reconstruction code. Next, identifying nearest neighbors in the reconstructed phase space and tracking their evolution over time using distance calculation functions like pdist2. Finally, estimating the Lyapunov exponent through linear regression of the average logarithmic divergence rate, which can be efficiently calculated using MATLAB's polyfit function for slope estimation. MATLAB's matrix computation capabilities are particularly suitable for such iterative calculations, especially when handling high-dimensional phase space reconstruction where vectorized operations can significantly improve performance.

It's important to note that practical applications require parameter adjustments based on specific systems, such as embedding dimension and time delay parameters, which directly affect result accuracy. These parameters can be optimized using MATLAB's optimization toolbox or through systematic parameter sweeping. For beginners, it's recommended to start algorithm validation with classical systems (like the Lorenz system) using predefined ODE solvers such as ode45, then gradually extend to real-world data analysis with proper normalization and validation procedures.