Calculating Embedding Dimension of Duffing Equation Using CAO Method with Code Implementation

Researchers can employ the CAO method to calculate the embedding dimension of the Duffing equation. The CAO method represents a computational technique for determining optimal embedding dimensions in nonlinear systems, particularly useful for analyzing chaotic dynamics. This computer-based approach projects original time series data into lower-dimensional spaces and systematically evaluates the embedding dimension through neighborhood analysis. Key implementation aspects include: - Phase space reconstruction using time-delay coordinates - Calculation of minimum embedding dimension through false nearest neighbors analysis - Adaptive thresholding to distinguish true neighbors from false ones The method effectively addresses limitations of traditional approaches by preventing overestimation or underestimation of embedding dimensions. Its algorithm typically involves: 1. Constructing delay vectors from the original time series 2. Computing distance ratios between neighboring points in progressively higher dimensions 3. Applying convergence criteria to identify optimal embedding dimension This robustness makes CAO method particularly valuable for investigating dynamical properties of nonlinear systems like the Duffing oscillator, where proper embedding dimension selection is crucial for accurate attractor reconstruction and subsequent analysis.