Lyapunov Exponent Computation Toolbox

Lyapunov exponents are essential metrics for quantifying the sensitivity of nonlinear dynamical systems to initial conditions and serve as key parameters for determining whether a system exhibits chaotic behavior. When the largest Lyapunov exponent is positive, the system demonstrates typical chaotic characteristics.

For classical chaotic models such as the Lorenz system, Chua system, and Lu system, specialized Lyapunov exponent computation toolboxes typically include the following core features:

Phase Space Reconstruction Using time-delay embedding methods to reconstruct observed one-dimensional time series into high-dimensional phase space, which is a fundamental step for analyzing experimental data. In code implementations, this often involves functions like delay_embed() that calculate optimal embedding parameters using false nearest neighbors or mutual information methods.

Jacobian Matrix Estimation For systems with known differential equations (e.g., Lorenz equations), the Jacobian matrix can be directly computed through symbolic differentiation. For scenarios with only observational data, methods like local linear fitting are employed for approximation, typically implemented using local_linear_fit() functions that perform regression on neighboring points in phase space.

Orthogonalization Algorithm Gram-Schmidt orthogonalization is periodically applied to adjust basis vector directions, preventing numerical issues where all exponents converge to the largest value during computation. This is commonly implemented through gram_schmidt() routines called at regular iteration intervals.

Exponent Convergence Determination By monitoring the stability of each exponent's temporal evolution, sufficient iteration steps are determined to ensure result reliability. Code implementations typically include convergence threshold checks and automatic iteration control using functions like check_convergence().

For discrete systems like Logistic maps, computational methods differ—directly using the limit of logarithmic growth rates derived from the product of mapping function derivatives. Such toolboxes typically distinguish processing pipelines for continuous systems and discrete mappings, with separate compute_continuous_lyapunov() and compute_discrete_lyapunov() functions.

Application scenarios include: stability analysis of circuit systems, validation of meteorological prediction models, and research on biological neural rhythms—all fields requiring quantification of system complexity. Notably, computation results are sensitive to noise, requiring combined filtering preprocessing in practical applications, often implemented through wavelet denoising or Kalman filter modules.