The central problem the researchers set out to address is a gap in how climate tipping points are usually studied. Earth System Models, the vast numerical frameworks that simulate the climate at fine spatial scales, do project tipping behaviour in future scenarios. Yet these models are computationally so demanding that they have been run over only a narrow range of emissions pathways. Very few simulations exist in which warming is deliberately reversed, the so-called overshoot scenarios in which temperatures rise past a threshold and then decline. That scarcity leaves scientists with a limited understanding of hysteresis, the phenomenon in which a system, once tipped, refuses to return to its original state even after the forcing that triggered the change has been undone. Without such understanding, policymakers hoping to stabilise the climate after a temporary overshoot cannot know which damages would be reversible and which would be locked in.

Huntingford and colleagues turned to the mathematics of nonlinear dynamical systems, which have been refined over decades and are well suited to describing abrupt changes of state. Their starting point is a cubic equation of a form long used to describe large-scale environmental systems, including the Atlantic Meridional Overturning Circulation. The equation contains a bifurcation parameter, a quantity that in this case is set by the amount of global warming since pre-industrial times. As warming increases, the parameter moves toward a fold bifurcation, the mathematical point at which the stable state the system has occupied disappears and the system is forced to jump to an alternative equilibrium. Crucially, if the parameter only briefly exceeds that point, tipping may be avoided, a behaviour that depends on the inertia of the system, its tendency to respond slowly over long timescales.

The genuine novelty of the new paper lies in the algebra that maps real, measurable attributes of a climate component onto the equation’s abstract parameters. The framework requires just five quantities: the level of global warming at which tipping occurs, the extent or magnitude of the system at the moment of tipping, the lower temperature at which hysteresis ends and the system can return to its earlier state, the system’s extent at that lower temperature, and a single parameter describing inertia. From these five user-defined values, the authors derive closed-form expressions for the four coefficients of the governing equation. This means that anyone with knowledge of a system’s tipping threshold and its hysteresis behaviour, whether drawn from palaeoclimate records or from complex model output, can calibrate the simple equation directly, without any fitting procedure.

To drive the model, the team constructed a warming trajectory that begins with the smoothed historical record of global temperatures from the NASA-GISS dataset, spanning 1880 to 2024, and then extends it smoothly into the future with a quadratic overshoot profile. The coefficients of that quadratic are constrained by the final value and the rate of change of the historical record, ensuring a seamless transition from observed warming to the idealised future, and by a user-chosen peak warming level. In their numerical example, the authors set a peak of 2.8 degrees Celsius above pre-industrial levels, well beyond the tipping threshold in their illustrative configuration, and then let temperatures decline back down. This construction allows the equation to be tested across the full arc of an overshoot: the approach to the threshold, the passage beyond it, and the long return.

The simulations reveal how decisively inertia shapes the outcome. With low inertia, the system tips as warming peaks, jumping to the alternative state and then tracing a full hysteresis loop as temperatures fall, only returning to its original branch once cooling drops below the lower fold. At intermediate inertia, the system makes an extensive excursion toward the new state but ultimately recovers, sliding back toward its initial condition without ever completing the jump. At very high inertia, the state variable barely moves at all. The physical intuition is straightforward: a sluggish system such as a massive ice sheet may simply not have time to respond before the forcing recedes, whereas a fast-responding system such as a coral reef, with little inertia, is far more vulnerable to a transient overshoot.

Going beyond numerical experiments, the authors performed a scale analysis by rewriting the equation in non-dimensional form. This transformation collapses the problem into a compact expression governed by three dimensionless parameter clusters that combine the system’s tipping attributes, its inertia, and the amplitude and curvature of the warming overshoot. From this form, the team recovered an inequality, consistent with earlier theoretical work by Ritchie and colleagues, that cleanly separates the cases in which an overshoot triggers full tipping and hysteresis from those in which the system escapes unscathed. For their illustrative parameters, the analysis shows that avoiding tipping requires the inertia parameter to exceed a critical value of roughly 27.8, in close agreement with the numerical simulations. The agreement between the analytical threshold and the computed trajectories is a satisfying validation of the framework.

The authors are careful about the limitations of their approach. The cubic structure of the equation, to some extent, predetermines the shape of the hysteresis loop, and real climate components may require perturbation terms or asymmetric basins of attraction. The model tracks a single state variable, whereas scientists worry that the activation of one tipping element could alter the timing of others, creating cascades of the kind explored in coupled network models. The framework also addresses only fold bifurcations, while tipping can also arise from oscillatory instabilities associated with Hopf bifurcations, or even without any bifurcation at all, through rate-induced tipping of the kind implicated in peatland fires. In cases where a simple equation cannot capture the dominant qualitative behaviour, the authors note that more complex models from the climate modelling hierarchy remain the appropriate tools.

Even with those caveats, the potential applications are considerable. Because the equation is computationally trivial to run, it can be forced across a far wider ensemble of warming pathways than any Earth System Model could ever explore, including the overshoot trajectories that are becoming central to climate policy debates. If the same framework is fitted to multiple Earth System Models, the resulting spread in parameter values offers a concise way to quantify why the models disagree about when tipping occurs, covering the warming threshold, the size of the jump, and the amount of cooling needed to escape hysteresis. There is also a path toward better early warning systems: adding high-frequency noise to the calibrated equation and studying how the system’s variability changes as tipping approaches could sharpen the statistical signals that researchers monitor in real climate data.

The timing of this work gives it particular resonance. Recent analyses suggest the planet may already be at or near the 1.5 degree threshold, meaning that any eventual stabilisation at that level is likely to occur only after a temporary overshoot, potentially enabled by carbon removal technologies. Whether such an overshoot is a survivable detour or a one-way door depends on the inertia and hysteresis characteristics of each tipping element, quantities that this new framework is designed to quantify from the evidence that already exists. By providing a complete, reproducible manual for mapping climate components onto a single dynamical equation, Huntingford and his colleagues have given the tipping points research community a tool that is simple enough to be used widely, yet rich enough to capture the full drama of a system that jumps, locks itself into a new state, and only lets go when the world has cooled far more than it warmed in the first place.

Subject of Research: A simple nonlinear dynamical system for representing climate tipping points, hysteresis and inertia under global warming overshoot scenarios

Article Title: A simple dynamical system for representing climate tipping points with hysteresis

Article References: Huntingford, C., Ritchie, P. D. L., & Clarke, J. (2026). A simple dynamical system for representing climate tipping points with hysteresis. Nonlinear Processes in Geophysics, 33(3), 385-399. https://doi.org/10.5194/npg-33-385-2026

Image Credits: AI Generated

DOI: 10.5194/npg-33-385-2026

Keywords: climate tipping points, hysteresis, dynamical systems, bifurcation, Earth System Models, global warming overshoot, AMOC, system inertia, nonlinear processes, early warning signals, Paris Agreement, fold bifurcation

Critical slowing down is a concept borrowed from nonlinear dynamics. When a system such as an ocean circulation sits far from a tipping point, it recovers quickly from small perturbations, and its natural fluctuations are fast and small. As the system approaches a bifurcation, the point at which it can no longer return to its current stable state, its recovery rate slows, and fluctuations become larger and more persistent. Statisticians track this process through indicators such as rising variance, rising lag-one autocorrelation, and a quantity called the restoring rate, which estimates how quickly the system snaps back toward equilibrium. In the extreme, these indicators can serve as early-warning signals of an impending transition, which is why they have become central to the debate over climate tipping points.

The AMOC is a prime candidate for such monitoring because it is thought to possess multiple stable states. The mechanism behind this bistability is the salt advection feedback: a weakening AMOC transports less salt northward, which reduces surface density in the regions where North Atlantic Deep Water forms, which in turn weakens the circulation further, potentially until an alternative, much weaker state is reached. Paleoclimate records, theoretical models and experiments with some general circulation models support the idea that such alternative states exist, although not all climate models reproduce them, and the question remains hotly debated. The Intergovernmental Panel on Climate Change concluded in its sixth assessment report, with medium confidence, that a collapse will not occur before 2100, a judgment resting heavily on the very models the new study scrutinizes.

Because continuous direct measurements of the AMOC’s strength have only existed since 2004, scientists studying longer timescales must rely on fingerprints, statistical patterns derived from observable variables that are physically connected to the circulation. The most widely used is the AMOC sea-surface temperature index, which averages sea-surface temperatures in the subpolar gyre region south of Greenland and subtracts the global mean. A landmark 2018 study used this index to argue that the AMOC has weakened over the twentieth century, and a 2021 analysis by Boers found a significant critical slowing down signal in temperature- and salinity-based fingerprints of the circulation. That signal has since been widely interpreted as evidence that the real-world AMOC is losing stability, and one 2023 study even used it to predict a tipping time, though other researchers have cautioned that the uncertainties in such predictions are enormous.

The problem is that the sea-surface temperature index is only partially correlated with the actual overturning circulation, and that correlation varies across models, time periods and scenarios. Other physical processes, such as changes in ocean mixed layer depth or atmospheric circulation, could in principle alter subpolar sea-surface temperatures in ways that mimic a stability loss without any real change in the AMOC. The researchers call this scenario a physical false positive: the statistical signal is genuinely present in the temperatures, but it says nothing about the circulation itself. Their central question was how confident we can be that the observed signal in the temperature index reflects the AMOC rather than such a confounding process.

To answer it, the team turned to the historical simulations of the sixth Climate Model Intercomparison Project, or CMIP6, in which 27 models ran the period 1850 to 2014 under realistic natural and anthropogenic forcings, with 133 ensemble members in total. For each run they computed three time series: the maximum overturning strength at 26.5 degrees north, the strength at 35 degrees north, and the sea-surface temperature index, yielding 399 AMOC time series. They then calculated three critical slowing down indicators for each, producing 1,197 indicator time series. Statistical significance was assessed conservatively, by generating 1,000 Fourier surrogates, random time series with the same variance and autocorrelation structure as the original, for each AMOC record, and asking how often a purely random process would produce an indicator trend as steep as the one observed.

The results were stark. Out of 133 ensemble members, only four showed a significant increase in the restoring rate at both streamfunction latitudes, and just one case showed coinciding significant increases in both the temperature index and a streamfunction strength. Eleven models, covering 96 ensemble members, showed no significant increases at all. Crucially, not a single model reproduced the pattern seen in the observations, where all three critical slowing down indicators rise significantly in the sea-surface temperature index. When the researchers counted all increases at the conventional 0.05 significance level, the numbers for the temperature index sat at or below what pure chance would predict for 133 random time series. In other words, the only statistically unusual thing about the model temperature indices is how few warning signals they contain.

The comparison between the temperature index and the direct circulation measure proved equally revealing. Because the models show no false warning signals in the temperature index despite the many other processes that could plausibly influence subpolar sea-surface temperatures, the researchers concluded that the index is not prone to physical false positives. If none of the 27 models contains any non-AMOC process capable of generating the observed signal, it becomes unlikely that such a process exists unrepresented in the real world. The logical implication is uncomfortable: the critical slowing down detected in real-world North Atlantic sea-surface temperatures is probably a genuine reflection of changes in the overturning circulation itself, quite possibly a loss of stability that the models simply fail to capture.

Why do the models diverge so sharply from reality? The authors lay out the possibilities. The model AMOCs may be genuinely stable while the real circulation is destabilizing, a difference that could stem from a well-documented bias in which climate models build in too much stability. Alternatively, the model AMOCs may be capable of tipping but start further from the tipping point than the real system, so their warning signals have not yet emerged from the noise. This second explanation gains support from known biases in the simulated freshwater convergence across the Atlantic basin, which is positive in most models and negative in observations, effectively placing the simulated circulation at a safer distance from the threshold at which tipping occurs. A handful of models, including CanESM5, CESM2, HadGEM3-GC31-LL and MIROC6, did show statistically unusual clusters of warning signals in some ensemble members, concentrated in the subpolar latitude, consistent with an AMOC approaching but not yet close to a tipping point.

The study carries practical lessons beyond the headline finding. It demonstrates that single ensemble members can show warning signals by chance, so robust conclusions about ocean stability demand many realizations per model, a point reinforced by evidence that the AMOC can respond differently to warming in different members of the same model. It also cautions that studies using the sea-surface temperature index as an emergent constraint should examine how higher-order statistics affect their results. Most importantly, the work sharpens rather than defuses the tipping point debate: by ruling out the most plausible alternative explanations for the observed signal, it strengthens the case that something is genuinely changing in the Atlantic overturning circulation, even as the models, biased toward stability, insist that everything is fine. The discrepancy between the simulated and the real Atlantic Ocean is now itself one of the most important open problems in climate science.

Subject of Research: Stability of the Atlantic Meridional Overturning Circulation assessed through critical slowing down indicators in CMIP6 historical simulations

Article Title: No critical slowing down in the Atlantic Overturning Circulation in historical CMIP6 simulations

Article References: Ben-Yami, M., Blaschke, L., Bathiany, S., & Boers, N. (2026). No critical slowing down in the Atlantic Overturning Circulation in historical CMIP6 simulations. PLOS Climate, 5(10), e0000781. https://doi.org/10.1371/journal.pclm.0000781

Image Credits: AI Generated

DOI: 10.1371/journal.pclm.0000781

Keywords: AMOC, critical slowing down, tipping points, CMIP6, climate models, North Atlantic, sea-surface temperature fingerprint, ocean circulation, early-warning signals, bistability, subpolar gyre, climate stability