Recurrence condensation during critical transitions in complex systems

Critical transitions in complex systems pose challenges for the healthy functioning of natural and engineered systems, sometimes with catastrophic outcomes. These critical points, where small changes cause large regime shifts, are difficult to detect—especially in noisy, high-dimensional settings. We investigate such a transition from chaotic to periodic oscillations via intermittency in a turbulent fluid mechanical system by using recurrence analysis. Recurrence plots (RPs) constructed from the time series of a state variable reveal a distinct progression from disordered, short broken diagonal lines to patches of ordered short diagonal lines and, ultimately, to a pattern of long continuous diagonal lines. This evolution in the recurrence patterns captures a transition from dynamics involving multiple time scales to a dominant single time scale; we term this phenomenon “recurrence condensation.” We quantify recurrence condensation using recurrence quantification measures, such as the recurrence time, determinism, entropy, laminarity, and trapping time, all of which show collapse to a single dominant time scale. Furthermore, these recurrence measures exhibit power-law scaling with the deviation of the control parameter from the critical point. Optimizing for the best power law reveals the critical value of the parameter. We apply this method to the synthetic data from a basic noisy Hopf bifurcation model and confirm that the detected critical point coincides with the bifurcation point. Our findings offer insights into identifying the critical points in noisy systems with gradual transitions, where the transition point is not well defined.