G-P Method in Chaotic Time Series Analysis

The G-P method (Grassberger-Procaccia method) in chaotic time series analysis is a classical technique for calculating correlation dimension and determining embedding dimension, particularly suitable for phase space reconstruction in nonlinear dynamical systems.

The core concept of the G-P method involves estimating a system's correlation dimension from time series data to reveal its intrinsic dynamic characteristics. The method first calculates the correlation integral between data points in the reconstructed phase space, then analyzes how the integral changes with distance variations, ultimately determining the correlation dimension. The magnitude of correlation dimension reflects the system's complexity level, while the choice of embedding dimension determines whether the phase space adequately unfolds the system's dynamic properties.

In practical implementation, the G-P method computes correlation dimension curves under different embedding dimensions and observes their convergence patterns to identify the optimal embedding dimension. This approach can be implemented using algorithms that calculate pairwise distances between phase space points, typically employing Euclidean distance metrics and Heaviside step functions for correlation integral computation. The method finds widespread applications in chaotic system analysis, signal processing, biomedical engineering, and other fields, providing an effective tool for studying nonlinear dynamic behaviors.