Calculating Lyapunov Exponents for Dynamical Systems

This program implements sophisticated numerical algorithms to calculate Lyapunov exponents for dynamical systems across different dimensions. The Lyapunov exponent serves as a fundamental quantitative measure of a system's sensitivity to initial conditions, playing a critical role in chaos theory, nonlinear dynamics, and physical system analysis. The core implementation involves: 1) Numerical integration of the system's equations to track trajectory evolution, 2) Calculation of the Jacobian matrix at each time step to characterize local linearized dynamics, 3) Orthonormalization procedures using QR decomposition to maintain numerical stability during exponent calculation, and 4) Long-term averaging of expansion/contraction rates along principal axes. Key functions include automated dimension detection, adaptive time-stepping for numerical precision, and multiple algorithm options (Wolf's method, Rosenstein's approach) for different system types. The program features a modular architecture allowing easy integration of user-defined dynamical systems through function handles. The intuitive interface provides clear parameter configuration, real-time progress monitoring, and comprehensive result visualization including exponent spectrum plots and convergence analysis. Designed for researchers, students, and professionals in mathematics, physics, engineering, and computational sciences, this tool offers robust numerical implementations with error handling and validation checks to ensure reliable results across various system complexities. The code includes extensive documentation with examples ranging from simple logistic maps to high-dimensional chaotic systems.