Climate tipping points represent critical thresholds, typically expressed as levels of global warming, beyond which positive feedbacks can cause components of the Earth system to irreversibly shift to a new state . Dansgaard–Oeschger events, observed in paleoclimate records, provide evidence of the capability of the Earth system to undergo such abrupt transitions . With arguably insufficient progress being made on reducing human reliance on burning of fossil fuels, crossing climate tipping points in the near future is an increasingly likely scenario. Multiple generations of sophisticated climate models have displayed tipping behaviour in future climate scenarios . Examples include, but are not limited to, dieback of the Amazon rainforest , collapse of large scale circulation currents in the Atlantic Ocean , melting of the polar ice sheets and alterations to atmospheric circulations .

The tipping behaviour observed in these complex climate models (or complex ESMs) may be amenable to characterisation and replication by low-dimensional nonlinear dynamical systems . Specifically, varying a parameter beyond a fold bifurcation can lead to bifurcation-induced tipping to an alternative state . However, if the parameter only temporarily exceeds a fold bifurcation, tipping of the system may not occur . The peak distance beyond the bifurcation and the duration of such an overshoot follow an inverse square law relationship, with the proportionality constant depending on the inertia of a system . A system with high inertia, such as large-scale ice sheets, is more likely to avoid tipping than a system with low inertia, such as coral reefs. This overshoot behaviour has been observed in both conceptual models and complex climate models . Policymakers are increasingly interested in considering the possibility of a climate overshoot , which corresponds to a temporary period of warming above key target thresholds, after which temperatures are lowered back to those levels, potentially using engineering solutions. The “Paris Target” aims for society to limit global warming to 1.5 °C above pre-industrial levels. However, the temperatures recorded in recent years suggest that the planet may be very close to exceeding that threshold , or may even be in the process of crossing it . Hence, any final stabilisation at the 1.5° threshold is likely to occur after a temporary overshoot in global warming.

However, a second key attribute of tipping points, should they be activated and a rapid move to a new state occur, is the potential for hysteresis. This effect arises when lowering the forcing that initially triggered a tipping point may not result in a return to the pre-tipping state, and instead, the system remains locked in the new state. Hence, if a temporary period of warming overshoot causes a tipping point to occur, the reduction of temperatures after a peak level and back to below the tipping point threshold may not lead to a reversion to the states observed when temperatures were increasing. Proposed examples of hysteresis in the climate system include the potential behaviour of the oceanic thermohaline circulation , large-scale ice sheets , the extent of mountain forest cover , the level of tropical forest cover at key locations and the position of the intertropical convergence zone . A broad summary of the climate attributes investigated for hysteresis, using one ESM that has been driven by a substantial ramp-up and ramp-down forcing in atmospheric CO2 and thus global warming, is presented in . However, some quantities, such as mean local rainfall in many locations, may show little hysteresis when mapped against a global temperature profile that reverses . Understanding hysteresis requires a comprehensive assessment of a system that can tip, extending beyond merely quantifying its behaviour at the time or threshold of tipping. Instead, a more wide-scale analysis is needed to determine the importance of hysteresis by identifying how much temperatures may need to decrease before a return to initial states is achieved. Hence, capturing all these factors likely requires dynamical system models with more degrees of freedom and parameters than are needed to investigate the tipping point alone.

Yet, although there are more complexities in understanding a system that can both tip and exhibit hysteresis when driven by forcings with reversal, much can still be gained from identifying dynamical system models that can capture such behaviours. One key benefit is that if the mapping of a full-complexity system onto a simpler structure is believed to be robust and captures the broad features of a nonlinear component of the Earth system, it may be forced for an extended set of pathways. This enables calculations for a much broader range of potential futures, which are not possible to perform with full Earth System Models (ESMs) due to their heavy computational requirements. This possibility is especially important for “overshoot” scenarios, as very few ESMs have been routinely forced for such trajectories. A second advantage is that valid simpler models capture overall behaviours, often through their bulk parameters. Should a single simpler model be fitted to a range of ESMs, the resulting different parameter values provide a potentially concise way to quantify inter-ESM differences. In the context of tipping points, this may involve characterising warming thresholds for tipping, the size of the jump to an alternative state, and the level of global cooling required to avoid hysteresis and return to any initial state. A third advantage is that if more high-frequency variability is added to the simpler model, and it is found to interact with nonlinearities in a way that mimics features of ESMs, this may open a path to creating better early warning systems for approaching tipping elements. Developing early warning systems for climate tipping points remains an area of active research and discussion . All of these advantages are possible if candidate dynamical system models are available for calibration against ESMs, both locally at tipping points and for their broader behaviours, including the period before tipping and any irreversibility (i.e. hysteresis) post-tipping.

Nonlinear mathematics has advanced substantially over recent decades e.g., yielding sets of equations with bifurcation parameters e.g.. Upon adjustment of such parameters, these equations often exhibit simulated changes of state and irreversibility. Considerable progress has been made in mapping features of the climate system that are expected to have the potential to tip onto dynamical systems equations. The emphasis thus far has focused on studying the period when the system is near or at the tipping threshold. Specifically, theory has been developed to understand whether, for some components of the Earth system, there is sufficient inertia to allow a temporary overshoot of a global warming level that, in equilibrium, would definitely cause tipping. Quantification of the possibility of inertia preventing tipping during a climate overshoot period has been undertaken for multiple parts of the Earth system .

To date, there has been far less progress in mapping the nonlinear behaviours of the Earth system onto dynamical systems when considering the full periods before and after tipping, or when tipping is averted due to inertia during temporary periods of high global warming. This omission is notable given the potential advances in understanding that could arise from such mappings. To bridge this gap, we present a relatively simple equation that exhibits tipping point and hysteresis effects. The particular novelty of our contribution lies in our explicit identification of key parameters that define a potential large-scale component of the climate system that exhibits such nonlinearity. The proposed system consists of five parameters: the level of global warming at which tipping occurs, the extent of the system at the point of tipping, a lower temperature representing the end of any hysteresis, the extent of the system at that lower temperature, and finally, a parameter defining inertia. A reader jumping to Eq. () will find a proposed dynamical system with the required properties. The equations shown before Eq. () allow a mapping of the parameters of the latter onto the five system-related parameters. Additionally, a reader can proceed to Eq. (), which outlines a forcing trajectory in global warming. This trajectory consists of the historical period, smoothly transitioning into an idealised future “upturned” quadratic pathway, with the ability to set the peak warming level.

To support reproducibility, we replicate every step of the algebraic derivations, acknowledging that this makes the manuscript lengthy. We hope that this approach enables the analysis to serve as a valuable manual on how, starting from scratch, one can map Earth system components onto a potential descriptive dynamical system equation. This equation can then be used to explore tipping points, hysteresis, and inertial effects, as well as a range of potential global warming pathways, both with and without overshoot.

Our aim is to provide an equation and a forcing profile capable of replicating a system that exhibits a tipping point, hysteresis behaviour, and inertia effects. We illustrate how the equation set may be calibrated (i.e. parameterised) and hope it will be useful to researchers wishing to quantify components of the Earth system believed to have strong nonlinear responses to climate change. Such beliefs may be driven by direct process understanding or based on assessments of projections by ESMs. We have aimed for a parameter-sparse model framework, but its development still requires substantial algebraic manipulations to illustrate how the main equations can be fitted to process knowledge. For full reproducibility, we have chosen to present all such algebra, though the majority of the background derivations are placed in the Appendix.

The paper leads to three main equations. Equation () describes the time-evolution of the Earth system component of interest, with the preceding algebra showing how to relate it to parameters that define an actual system. Equation () presents a representative future “overshoot” forcing profile in temperature, with the preceding algebra relating it to a prescribed maximum level of global warming and ensuring a smooth transition onwards from the historical record of global warming. Equation () provides a condition that differentiates whether, for the parameters provided, a system exhibits tipping and hysteresis, or if the warming trajectory instead avoids a change of state.

The risk of triggering tipping points in major parts of the Earth system remains a key concern regarding potential responses to a changing climate as atmospheric greenhouse gases increase. Tipping refers to a shift of an Earth system component to a new state, which may occur with relatively little additional climate change. This new state may also exhibit hysteresis, implying that any attempt to reverse climate change after tipping might not restore the system to its original state. Research centres have contributed Earth System Models (ESMs) to the Climate Model Intercomparison Project version 6 , which inform the IPCC reports . ESMs predict various tipping points, but there is little consensus among these models about the level of climate change that would trigger their activation. Moreover, specific tipping points do not necessarily appear in all ESMs. To characterise the simulated tipping points, they should ideally be mapped onto a common equation framework, with parameters aligned to each ESM. Such equations may also enable the investigation of additional possible climate change trajectories for which ESMs have not been operated.

We begin with a moderately basic nonlinear ordinary differential equation, Eq. (), which exhibits tipping point and hysteresis effects. This equation, of “cubic” form, has been proposed by many to describe features of the Earth system (see Sect.  for references), but there is presently no routine method to describe how to map climate tipping point features onto this model. To do this, we use a variable X to represent our system state, denoting a component of the Earth system of interest. We assume that four quantities are available to define the background equilibrium states, either already known (e.g. estimated from palaeo data) or obtained from more complex ESM calculations. These parameters include the value of X at tipping, XTP, and the global warming temperature at which it occurs, ΔTTP. The other two parameters are the value of X at the end of hysteresis, XHYS, and the corresponding lower global warming temperature at which it occurs, ΔTHYS. These four parameters are illustrated in Fig. .

The particular form of the governing equation that we propose is our key process Eq. (). If we initially neglect the transient left-hand-side term (by setting it to zero), the remaining equation describes the pseudo-equilibrium solutions for various values of global warming, ΔT. In this context, the assumption is that there are very slow palaeo-timescale changes in ΔT, which we consider as a bifurcation parameter. The right-hand side of Eq. () involves four unknown parameters, β,μ0,a0 and a1. These parameters are functions of our four user-defined parameters, XTP,ΔTTP,XHYS and ΔTHYS, as specified by Eqs. ()–().

We introduce transient effects into our calculations in two ways. We assume that the system simulated by the variable X has inertial effects, which we represent by a fifth user-defined parameter, cp, in Eq. (). Then we introduce a time-evolving profile for global warming, ΔT(t), which combines the historical record with a future “overshoot” scenario that we model as quadratic. The three quadratic coefficients are partially constrained by the magnitude (ΔT+) and the rate of change of warming (δ) for the contemporary period, thereby ensuring a smooth transition from the warming of the recent past to that of the simulated future. The required third constraint is a user-prescribed value for the peak level of global warming, ΔTP. This warming profile leads to our second key equation, Eq. (), as illustrated in Fig.  and for our value of ΔTP=2.8. The three parameters of ΔT+,δ and ΔTP are translated into the three parameters e,f and g, given by Eqs. (), () and () respectively.

Our second aim is to investigate and characterise solutions to Eqs. () and () for a range of inertial values, cp, in response to human-induced climate change. The focus on an overshoot warming profile is deliberate, as it allows us to consider the potential for hysteresis should a tipping point be activated and there is a reversal of forcing. If society achieves eventual climate stabilisation at a global warming level of 1.5 or 2 °C above pre-industrial levels, this may occur after a temporary period of higher global temperatures. Such an overshoot is possible, given that current emissions levels may not be falling sufficiently fast to allow the climate system to approach a stabilisation maximum temperature only from below. Using numerical calculations, we find that for high inertia, the variable X will not exhibit major tipping and hysteresis (Fig. ). Employing the nondimensionalisation method to our model and forcing equations, we estimate with Eq. () a threshold that determines whether tipping can be avoided. These calculations relate to existing analysis e.g. that estimates the risk of tipping when the equilibrium level for activation is temporarily crossed. If there is additional understanding leading to a known value of cp, then the model equation, Eq. (), is fully constrained by process knowledge.

Equation () is available to assess responses to forcings of alternative future trajectories in global warming, ΔT. Our quadratic form, which we consider as eventually returning to pre-industrial conditions, may be replaced with more sophisticated profiles that stabilise at key warming thresholds, either with or without a period of overshoot. Algebraic forms exist for such profiles e.g., opening the possibility of further analytic assessment, again without relying solely on numerical understanding. The equation system is also available for forcing with monotonically increasing global temperatures, which is likely to occur during the 21st Century if society follows any of the standard Shared Socioeconomic Pathways (SSPs) that are associated with the continued substantial burning of fossil fuels.

Due to concerns that human-induced climate alteration may cause key components of the Earth system to tip, there is substantial interest in developing early warning systems for such events e.g.. Although generic types of such warning systems exist, such as those that assess for evidence of critical slowing down, more bespoke forms may be developed for particular parts of the climate system. We suggest that a calibrated dynamical system, which accurately describes a key process, could also support the development of such warning systems. For instance, adding high-frequency noise to Eq. (), either to ΔT or X, and studying the simulated responses and how they change as tipping approaches, may lead to the discovery of more accurate warning methods.

Our analysis has some caveats. The generic cubic form, in X, of the right-hand side of Eq. (), to some extent, predetermines any path taken by the state variable in the presence of hysteresis. The actual components of the climate system may exhibit slightly different forms, necessitating more complex algebraic expressions that introduce perturbation terms to the cubic structure. In addition, utilising the normal form and a linear change in variables assumes a certain symmetry to the problem, for example, as seen in Fig. . At the midpoint between the folds of ΔTHYS and ΔTTP the distances to the basin boundary (the middle unstable branch), from the stable top and lower branches, are the same. However, some problems might require a nonlinear coordinate change, which could lead to asymmetric basins of attraction. The equations we provide model only a single state variable, X, while there is concern that the activation of a tipping point may alter the timing of others, creating a cascade effect. A suggested set of equations that captures such effects, along with their couplings, is provided by . In our modelling framework, we only consider the passage through fold bifurcations. However, tipping can arise from oscillatory instabilities, which are often attributed to passing through a Hopf bifurcation. Or, alternatively, tipping can even occur without crossing any bifurcations via rate-induced tipping, such as those attributed to peatland fires .

In summary, we provide a relatively simple equation on to which the behaviour of climate components expected to exhibit a tipping point, leading to a new state, can potentially be mapped. The equation can simulate hysteresis effects. Given the interest in the risk of climate tipping points being triggered by further human-induced climate change, and noting that ESMs differ substantially in their projections of the timing of such events, these mappings may offer a direct method to compare climate projections. The key equation is Eq. (), and its parameters are given by Eqs. ()–(). Simplified equations that effectively emulate the bulk features of complex components of the climate system are suitable for investigating a broad range of future climate change scenarios, which may not be feasible with ESMs due to their high computational cost. We note that some climate research centres are starting to force ESMs with overshoot-type scenarios, and any simulated hysteresis effects may be amenable to mapping onto our dynamical system framework. Here, we examine overshoot scenarios using the key profile of Eq. (), with its parameters given by Eqs. ()–(). We utilise this framework to explore how the prescription of system inertia determines whether full tipping with hysteresis occurs, and we link our findings analytically back to the existing literature via Eq. (). There are already extensive and valuable analyses of potential tipping points in the literature, focusing on behaviour at the time when a major change in a climate component might be expected. Such literature includes examples using the “cubic” form (see Sect. ), while also noting in particular the historical development of dynamical systems to represent AMOC changes e.g.. We hope that the equations presented here provide an opportunity to more fully quantify behaviours over longer periods, both before and after potential climate tipping events.