The complex dynamics of correlations within chaotic systems

It is widely reported that the functional connectivity estimated by statistical correlations is often varying within nonlinear systems. Generally, these varying correlations between time series are detected by sliding windows. Still, it is unclear how these correlations evolve within a chaotic system. This work intends to give a quantitative framework to identify the dynamics of correlations within chaotic systems. To this end, we embed the pairwise statistical correlations (from time series within a system) into a correlation-based system by sliding windows. This allows for detecting the dynamics of correlations within a complex system through the embedded correlation-based system. Three chaotic systems (i.e., the Lorenz, the Rossler, and the Chen systems) are employed as benchmark examples. We find that both linear and nonlinear correlations within three chaotic systems show chaotic behaviors on some short window sizes, then transit to non-chaotic states with window size increasing. Moreover, the chaotic dynamics of nonlinear correlations exhibit higher uncertainty than the linear one and the original chaotic systems. The chaotic behaviors of correlations within chaotic systems give another evidence of the difficulty of prediction for chaotic systems. Meanwhile, the identified state transitions (concerning the window size) of correlations may provide a quantitative rule to select an appropriate window size for sliding windows.