MATLAB Implementation for Calculating Lyapunov Exponents of Various Chaotic Systems

This article demonstrates how to calculate Lyapunov exponents for various chaotic systems using MATLAB. Lyapunov exponents serve as quantitative measures of chaotic behavior, where positive values indicate exponential divergence of nearby trajectories - a fundamental characteristic of chaotic systems. The implementation typically involves numerical integration of system equations combined with orthogonalization procedures (e.g., using Gram-Schmidt method) to track the evolution of perturbation vectors. Key computational aspects include: - Employing ODE solvers (ode45, ode15s) for system trajectory integration - Implementing linearized system Jacobians to compute local divergence rates - Applying QR decomposition at regular intervals to maintain orthogonality of deviation vectors - Accumulating growth rates over sufficient iterations for statistical reliability Users can extend functionality by incorporating custom chaotic system definitions through function handles, modifying parameters for sensitivity analysis, or implementing alternative algorithms like Wolf's method for continuous systems. While requiring foundational knowledge in nonlinear dynamics and numerical methods, mastering these techniques enables deeper exploration of chaotic system properties, including predictability limits and stability characteristics. The provided framework supports comparative analysis across different chaotic systems like Lorenz, Rössler, and Chua circuits.