Small Data Method for Calculating Lyapunov Exponents

To compute Lyapunov exponents using small datasets, a mixed programming methodology can be employed. This approach integrates multiple programming languages such as Python for data preprocessing and visualization, and C++ for high-performance numerical computations. The algorithm implementation typically involves phase space reconstruction using time-delay embedding techniques, followed by nearest neighbor search algorithms to track trajectory divergence. Additionally, signal processing methods including wavelet analysis for multi-resolution decomposition and Fourier transform for frequency domain analysis can be incorporated to extract maximum information from limited data. Key computational steps may involve Jacobian matrix estimation through local linear fits and exponential growth rate calculation using robust regression techniques. This hybrid implementation ensures computational efficiency while maintaining accuracy, enabling reliable Lyapunov exponent estimation even with constrained data availability.