One Equation to Map Climate Tipping Points and Their Reversibility

Climate scientists have long warned that parts of the Earth system, from the Atlantic Ocean’s overturning circulation to the Amazon rainforest and the great polar ice sheets, may not respond to global warming in a smooth, gradual way. Instead, they may pass critical thresholds beyond which relatively small additional warming triggers an abrupt jump into a new state. A new study published in the journal Nonlinear Processes in Geophysics by Chris Huntingford of the UK Centre for Ecology and Hydrology, together with Paul D. L. Ritchie and Joseph Clarke of the University of Exeter, offers a deceptively simple mathematical tool for capturing not only those jumps but also what happens afterwards, when temperatures are brought back down. The work arrives at a moment when the world is edging uncomfortably close to the 1.5 degrees Celsius warming limit set by the Paris Agreement, making the question of what happens during and after a temporary overshoot of that threshold urgently practical.

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