Determining Embedding Dimension for Rossler System Using Cao's Method with Time Delay from Mutual Information

In nonlinear time series analysis, determining appropriate time delay and embedding dimension are critical steps for phase space reconstruction. For chaotic systems like the Rossler system, proper parameter selection effectively preserves the system's dynamic characteristics.

Mutual Information Method for Time Delay The mutual information method calculates mutual information between time series at different time delays, selecting the first local minimum as the optimal delay. This approach is more suitable for nonlinear systems than traditional autocorrelation functions because it captures nonlinear dependencies. Implementation typically involves creating delayed versions of the time series and computing mutual information using histogram-based estimation or kernel density methods. The algorithm searches for the first significant minimum in the mutual information curve, which corresponds to the point where the delayed series provides minimal redundant information.

Cao's Method for Embedding Dimension Cao's method is an improved approach based on false nearest neighbors (FNN) that automatically determines the minimum embedding dimension by calculating the change rate of neighboring points as the embedding dimension increases. Its advantage lies in avoiding subjective threshold settings, making it suitable for complex systems like the Rossler attractor. The method works by tracking how the distance between nearest neighbors changes when increasing the embedding dimension from m to m+1. When the percentage of false nearest neighbors stabilizes below a small threshold, the current dimension m is considered sufficient. The key function calculates the E1(m) and E2(m) parameters, where E2(m) becomes constant at the proper embedding dimension.

Combined Application First, obtain time delay τ using the mutual information method, then use this τ value as input for Cao's method. Gradually increase the embedding dimension m until the proportion of false nearest neighbors stabilizes. The corresponding m value represents the minimum embedding dimension for the Rossler system. This combined approach enables more accurate phase space reconstruction, laying the foundation for subsequent dynamic analysis such as Lyapunov exponent calculation. Code implementation would typically involve: 1) Preprocessing the Rossler time series, 2) Applying mutual information algorithm to find optimal τ, 3) Using τ in Cao's method with incremental m values, 4) Monitoring the FNN ratio until convergence is achieved.