[DSRP Evidence](https://dsrpevidence.org/) # A simple dynamical system for representing climate tipping points with hysteresis ## Details **Authors** Chris Huntingford, Paul D. L. Ritchie, Joseph Clarke **Year** 2026 **Publisher** Nonlinear Processes in Geophysics **Kind of work** article **Discipline** Climatology **Secondary disciplines** Mathematics **Applied** false [Read it at the publisher](https://doi.org/10.5194/npg-33-385-2026) 10.5194/npg-33-385-2026 ## In authors' words ### Abstract The risk that the climate system may contain tipping points remains a concern. Firstly, a level of global warming may be reached at which relatively small additional warming could cause major parts of the Earth system to transition to a new state. Depending on the location and specific Earth system component, this could disproportionately impact large sectors of society. Secondly, the Earth system component may exhibit hysteresis effects, and therefore, if global temperatures are subsequently lowered after triggering a jump in state, a return to earlier conditions may not occur until warming is substantially reduced. Earth System Models (ESMs) are numerical frameworks that operate at fine spatial scales. Such models are designed to estimate how all components of the climate system will evolve in response to changes in atmospheric greenhouse gas concentrations caused by human activity. Many ESM projections suggest that various parts of the climate system are capable of tipping. Yet, these models are computationally demanding and have therefore been operated only over a small range of scenarios. Very few "overshoot" simulations with ESMs exist, where climate change is reversed, resulting in limited understanding of hysteresis effects following a tipping event. Advances in nonlinear mathematics include developing dynamical systems whose equations frequently contain a bifurcation parameter. These equations can accurately replicate tipping points, jumps in state, and hysteresis as the bifurcation parameter changes. Recent progress in ESM development often introduces higher spatial resolution, enhancing process representation and increasing accuracy in predicting local changes. However, mapping the broad behaviour of the components of ESM projections onto simpler dynamical system models may also offer many advantages, including the overall characterisation of climate models and a method to rapidly extrapolate their projections to a wider range of forcing scenarios. The bifurcation parameters in such equations may represent changing forcings, such as an increasing warming level that leads to a tipping event. Progress has already been made in mapping many components of the Earth system onto large-scale variables for representation as dynamical systems. Most advances to date have focused on understanding whether tipping events can be avoided if systems possess substantial inertia, i.e. respond over long timescales, allowing climate change to temporarily exceed thresholds that might otherwise trigger major nonlinear change. However, potential hysteresis effects in the context of climate change are less well represented in equation form. Achieving such a mathematical formulation requires a dynamic system to describe a climate system component not only at the point of tipping but also for substantial periods before and after. This behaviour corresponds to a bifurcation parameter that first increases and then decreases, with the modelled behaviours differing significantly during the return phase. To support such necessary developments, we present a parameter-sparse dynamical system model that can exhibit hysteresis following a tipping occurrence, offering the potential for characterising Earth system components with this feature. We place particular emphasis on presenting in full the algebra needed to map known or modelled key attributes of a system that can tip onto the simplified dynamical system equation. We drive the equation with a time-evolving forcing representing an "overshoot" trajectory of global warming that exceeds a threshold for potential tipping. Calculations are performed over a range of system inertia values, illustrating a threshold inertia above which full tipping and hysteresis can be avoided. We use scale analysis to relate this threshold to those reported in existing climate research on behaviour near potential tipping points. In some instances, a basic equation may be too simple to capture the dominant qualitative be ### What they set out to do (purpose) To develop a mathematically simple dynamical-system model that captures both the sudden state change ("tipping") and the hysteresis that can follow it in components of the Earth's climate system, as a fast alternative to full Earth System Models. ### Who or what was studied (sample) Not an empirical sample; the study develops and analyzes a five-parameter dynamical-system equation representing generic tippable Earth-system components under a warming-then-cooling ("overshoot") forcing trajectory. ### How they did it (methods) Mathematical modeling: derivation of a bifurcation-parameter equation, algebraic mapping of Earth-system attributes onto the model, and calculation of system behavior across a range of inertia values under an overshoot warming scenario, compared against known climate-tipping thresholds using scale analysis. ### What they found (results) The five-parameter model shows that a climate-system component driven past a tipping threshold and then back down avoids permanent state change only above a threshold level of system inertia, below which the return path differs systematically from the path that caused tipping, producing hysteresis. ## Commentary ### In short The finding shows that whether a system crosses a threshold into a new state depends on a directional, history-dependent relationship between rising and falling forcing (hysteresis) together with the system's own part/whole properties such as inertia and extent, illustrating distinctions, relationships, and systems together. **Patterns it shows** D, S, R **Added** 2026-09-26 **How to cite this** Chris Huntingford, Paul D. L. Ritchie, Joseph Clarke (2026). A simple dynamical system for representing climate tipping points with hysteresis. Nonlinear Processes in Geophysics.