Computing Chaotic Parameters and Phase Space Reconstruction Parameters Using MATLAB

In nonlinear dynamics analysis, chaotic parameters and phase space reconstruction are essential tools for understanding complex system behaviors. MATLAB provides robust computational and visualization capabilities that facilitate efficient implementation of these analyses.

Chaotic Parameter Calculation Common chaotic parameters include Lyapunov exponents, correlation dimension, and Kolmogorov entropy. These parameters help quantify the chaotic characteristics of systems. For instance, Lyapunov exponents measure a system's sensitivity to initial conditions—positive exponents typically indicate chaotic behavior. In MATLAB, these parameters can be computed through numerical methods such as fitting differential equations using time series data or directly processing observational data. Implementation often involves using functions like `lyapunovExponent` for calculating maximal Lyapunov exponents from time series, where algorithms like Rosenstein's method or Wolf's method can be applied.

Phase Space Reconstruction Phase space reconstruction transforms one-dimensional time series into higher-dimensional phase space to reveal underlying dynamical properties. Key parameters include embedding dimension (m) and time delay (τ). Common methodologies include: - Autocorrelation or mutual information methods: Used to determine optimal time delay τ, where MATLAB's `autocorr` function or custom mutual information calculations can be implemented. - False Nearest Neighbors (FNN) method: Employed to identify appropriate embedding dimension m, achievable through algorithms that minimize false neighbor percentages across increasing dimensions.

MATLAB Implementation MATLAB's Signal Processing and Nonlinear Dynamics toolboxes (e.g., the `phaseSpaceReconstruction` function) assist in these computations. Users can also develop custom algorithms, such as applying Takens' theorem for phase space reconstruction and visualizing system attractor structures through phase plots. For example, reconstructing phase space using `delayEmbed` with optimized τ and m parameters allows plotting attractors via `plot3` or `scatter3` functions to observe topological properties.

By appropriately selecting parameters and algorithms, MATLAB enables effective analysis of chaotic system characteristics, providing foundations for modeling and predicting complex systems.