Code Implementation for Computing Important Parameters in Chaotic Systems Including Lyapunov Exponents

The code has been enhanced with comprehensive modules for computing chaotic system parameters, making it more complete. The calculation of Lyapunov exponents represents a crucial parameter computation in chaotic systems. When implementing Lyapunov exponent calculation, the code first analyzes the system's evolutionary trajectory using phase space reconstruction techniques, then performs the computation based on the analysis results through numerical differentiation and eigenvalue decomposition algorithms. Beyond Lyapunov exponents, numerous other important parameters require computation in chaotic systems, such as attractor dimension (calculated using correlation dimension or box-counting methods) and bifurcation parameters (determined through parameter continuation algorithms). Therefore, integrating these parameter calculation modules into the code makes it more comprehensive for chaotic system analysis. Key implementation aspects include: - Phase space reconstruction using time-delay embedding methods - Numerical computation of Lyapunov exponents through Jacobian matrix estimation - Correlation dimension calculation using Grassberger-Procaccia algorithm - Bifurcation analysis through parameter sweeping and stability detection The code structure allows modular integration of these algorithms while maintaining computational efficiency and accuracy.