Tipping (or sudden transition) from a desirable state to an undesirable one can result in catastrophic changes, affecting natural environments, human societies and economies. Early warning signals (EWSs) are developed to forewarn such an impending tipping. However, when control parameters are varied fast, we observe that EWSs detect an impending tipping past bifurcation points; this questions the applicability of EWSs in preventing tipping. At the same time, the fast rate of change of parameter delays the actual tipping event, providing a borrowed window of safe operation. This delay in tipping offers a window for prevention through swift action. We demonstrate instances of both successful and unsuccessful preventions, in a paradigmatic thermoacoustic system-a horizontal Rijke tube. This work highlights the interplay of warning time, choice of prevention action and rate of variation of parameters in EWS-based prevention of tipping.
This paper critically examines the practical applicability of Early Warning Signals (EWSs) for preventing critical transitions, particularly when control parameters change rapidly. A significant finding is the demonstration that under such rapid changes, EWSs can alert after the system has already crossed its bifurcation point, thus rendering traditional prevention strategies ineffective. This challenges the conventional understanding of EWSs as purely preventive tools, highlighting a crucial limitation in real-world systems where parameters often fluctuate at finite rates.
The key innovation lies in moving beyond prevention to propose and experimentally validate EWS-based control actions for mitigating critical transitions after tipping has occurred. Specifically, the authors show that when prevention fails due to fast parameter variation, abruptly reducing the control parameter below the fold point (not just into the bistable region) can successfully return the system to its desirable, quiescent state. This introduces a new paradigm for managing critical transitions: using EWS insights not only for pre-emptive action but also for guiding post-tipping recovery, especially in systems that can tolerate temporary excursions into an undesirable state. The study meticulously details the failure mechanisms of preventive strategies—such as the “freeze strategy” (stopping parameter change upon EWS alert) and “direct cut-off” (reducing parameter to within the bistable region)—attributing failures to delayed EWS alerts and the “borrowed stability” that delays the actual tipping when parameters change quickly.
The main prior ingredients informing this research are foundational concepts in dynamical systems and critical transitions. These include:
1. Bifurcation Theory: The paper relies heavily on the understanding of bifurcation points (specifically Hopf and fold bifurcations) and hysteresis, which describe the sudden qualitative changes in system behavior as control parameters are varied. The subcritical Hopf bifurcation, leading to thermoacoustic instability, is the specific tipping phenomenon investigated.
2. Critical Slowing Down (CSD): This is the underlying principle for many EWSs, where the system’s recovery rate from perturbations slows down as it approaches a tipping point. EWSs like variance and autocorrelation, derived from CSD, are implicitly or explicitly built upon here.
3. Rate-Dependent Tipping/Delay: Prior research established that the finite rate of parameter change can delay the actual tipping beyond the static bifurcation point, a phenomenon termed “borrowed stability.” This concept is central to explaining why EWS alerts might be received too late for prevention.
4. Early Warning Signals (EWSs) Methodology: The paper uses the Hurst exponent (H) calculated via multifractal detrended fluctuation analysis (MFDFA) as its specific EWS. This statistical tool quantifies the fractal characteristics of time series, providing an indicator of approaching instability. The paper also leverages methods for evaluating EWS performance, such as Receiver Operating Characteristic (ROC) curves.
5. Thermoacoustic Instability (TAI): The experimental platform, a horizontal Rijke tube, is a well-studied system known for its transitions to TAI, which serves as a realistic and accessible model for critical transitions in engineering systems (e.g., combustion systems). The specific dynamics of pressure fluctuations and their relation to control parameters in this system are crucial for the experimental setup and observations.