A simple dynamical system for representing  climate tipping points with hysteresis

Huntingford, Chris; Ritchie, Paul D. L.; Clarke, Joseph

doi:10.5194/npg-33-385-2026

Articles | Volume 33, issue 3

https://doi.org/10.5194/npg-33-385-2026

© Author(s) 2026. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/npg-33-385-2026

© Author(s) 2026. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 33, issue 3

Research article

|

06 Aug 2026

Research article |

| 06 Aug 2026

A simple dynamical system for representing climate tipping points with hysteresis

Chris Huntingford, Paul D. L. Ritchie, and Joseph Clarke

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Final revised paper (published on 06 Aug 2026)
Preprint (discussion started on 10 Feb 2026)

Interactive discussion

Status: closed

RC1: 'Comment on egusphere-2026-550', Anonymous Referee #1, 02 Mar 2026

Comments in the PDF above

Citation: https://doi.org/10.5194/egusphere-2026-550-RC1
RC2: 'Comment on egusphere-2026-550', Anonymous Referee #2, 10 Mar 2026

The authors present a study of a simplified dynamical system displaying tipping points, with the claim that it can be calibrated to represent some aspects of the behaviour of more complex Earth Systems Models under global warming. The explicit aim is to include effects of 'inertia' and 'hysteresis'. Only one of the hysteresis branches is explored (the return path to the base state is not considered).
The paper is written more as the lecture notes of an elementary course on dynamics and bifurcations than as a research paper. The main result is in Fig. 4: fast crossing of a tipping point and return back of a control parameter does not trigger a jump in the system state, in contrast with the behavior for a slow crossing. This result is very well known. In fact, the authors check that the time scales agree with the result of Ritchie et al 2019, thus confirming that the behaviour has been studied in detail before. Thus, the only part that can be useful for the community is the parametrization of this simplified model in terms of quantities that in principle can be measured from a larger climate model. The manipulations leading to this parametrization are elementary, and do not deserve the many pages devoted to it. But there are more important caveats:
- The manuscript model, Eq. (1) is just the normal form of the codimension-2 cusp bifurcation. It can be argued that any dynamical system with only fold bifurcations can be generically brought to that form. But in general the change in variables needed for that is nonlinear, not the simple linear scaling used here for mu and X. The consequence is that the basins of attraction of the upper and lower stable-solution branches would be quite asymmetric, an important fact neglected in the present approach.
- Only fold bifurcations are taken into account. This excludes possibilities already observed in existing climatic models, such as oscillatory instabilities, homoclinic bifurcations, crisis and other global bifurcations, etc.
Anyway, I would accept that all the above criticisms would become irrelevant if the authors would be able to demonstrate their main claim: that this approach allows to map different complex climate models into a single framework, thus allowing comparison among them. The authors have not demonstrated this, and it seems difficult to do since, as recognized by the authors, most existing studies only consider short simulations not always exploring beyond a bifurcation point. I invite the authors to really follow their proposal and compare outputs from different complex models to show really that their approach is powerful (and in this case the straightforwardness of the calculations will be a welcomed feature). Until they do something like this I do not find the manuscript relevant to the community and I can not recommend the paper for publication.

Citation: https://doi.org/10.5194/egusphere-2026-550-RC2
AC1: 'Comment on egusphere-2026-550', Chris Huntingford, 12 May 2026

We thank the two reviewers for their help. Please find attached our proposed responses to their requests.

Citation: https://doi.org/10.5194/egusphere-2026-550-AC1

Peer review completion

AR – Author's response | RR – Referee report | ED – Editor decision | EF – Editorial file upload

AR by Chris Huntingford on behalf of the Authors (08 Jun 2026) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (11 Jun 2026) by Vicente Perez-Munuzuri

RR by Anonymous Referee #1 (11 Jun 2026)

RR by Anonymous Referee #3 (15 Jun 2026)

Suggestions for revision or reasons for rejection

The authors describe a generic dynamical system that can be used to model climate tipping elements and their hysteresis properties. More concretely, they formulate a third-order ordinary differential equation (ODE) as the minimal system to exhibit all desired properties, and subsequently derive relationships between the ODE parameters and relevant system characteristics. They analytically investigate the model's safe parameter regions, both in the context of bifurcation-tipping and overshoot scenarios. The manuscript is highly instructive and well written. It establishes a full manual for the application of this model in the context of climate tipping points and can serve as a basis for many existing studies that employ the heuristic concept of bifurcations for tipping points.

I have one major problem with the current presentation though. Throughout the manuscript it sounds as if this ODE is a novel model for climate tipping points with hysteresis. This is simply not the case and caused me to be very skeptical of the manuscript at first. The mathematical derivations and in-depth description of the model properties from first principles warrant publication, but the manuscript should give a history of the prior usage of this model, rather than present it as a novel development.

The authors briefly cite one important prior contribution already (10.5194/esd-12-69-2021). It should be made clear how the current manuscript and its results relate to that work. Beyond this, I found the following publications which use the discussed model for climate tipping point hysteresis and there may be others:
- Berglund and Gentz, 2002, Metastability in simple climate models: Pathwise analysis of slowly driven Langevin equations
- Dakos and Kefi, 2022, Ecological resilience: what to measure and how
- Boers et al., 2022, Theoretical and paleoclimatic evidence for abrupt transitions in the Earth system
Hystorically, Cessi, 1994, should also be mentioned as an early occurrence of this model in climate science.

Apart from this, there are the following minor points to address.
- The framework silently assumes X_HYS > X_TP, i.e. that the state variable increases upon tipping. The orientation convention should be stated explicitly and caveated with the fact that many natural systems like the AMOC tip downwards. I assume all the derivations carry over symmetrically but it is still good to mention this I think.
- The Ritchie et al. (2019) criterion behind Eq. (24) is asymptotic for small overshoots near the fold. Here the overshoot (0.5 K) is not small relative to the hysteresis width (1.9 K), so the agreement with Fig. 4 is empirical rather than guaranteed, this might be worth a sentence.
- The baseline definition of ε is given three different ways: "1880–1900 inclusive" (text), t = 0 to 21 (formula under Eq. 8), and "1880 to 1899 inclusive" (Fig. 3 caption), please reconcile.
- Fig. 1 caption: "β, µ_0, a_1 and a_1", last should be a_0.
- Abstract: "a method extrapolate their projections", missing "to".
- Eq. (22): stray comma before the closing bracket.
- Eq. (8): the ε line is an unnumbered continuation ending in a dangling comma.
- Eq. (25): use ≈ rather than = for rounded values.
- Fig. 3 annotation uses ∆T_p (lowercase) vs. ∆T_P in text.

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RR by Anonymous Referee #4 (06 Jul 2026)

Suggestions for revision or reasons for rejection

This manuscript presents a one equation box model, with a detailed mathematical analysis of this model, that is intended to represent the role of inertia in climate tipping point problems. The manuscript is well written, if a bit detail oriented, and has some contact with climate literature, though much of this contact is ahistorical.

The tone of the abstract is not appropriate for a technical journal. I don’t think someone who makes it a habit to read NPG needs to have the concept of a dynamical system to be “defined” in the public talk sense in the abstract.

There are multiple statements in the abstract I would disagree with as a climate scientist, and this will be revisited below. For example, it is the view of most climate scientists that ESMs are in fact not finely resolved enough. There is a role for many kinds of climate models (GCMs, EMICs, box models) with progress based on a combination of modelling strategies.

If the intent of the Introduction is to provide context, I would suggest focusing on at least a couple of areas and quoting papers over longer time scales. There is work on cessation of the AMOC, for example, that has a decades long history (it even led to a Nobel prize).

The model discussed is a first order, nonlinear differential equation. I like simple models, but the possible dynamics of a system like this is extremely limited (and if one goes back to Einstein there caution should be exercised about models that are too simple). This is textbook dynamical systems theory, and indeed while long series of calculations are presented nothing in the manuscript provides an indication that the dynamics represent the true Earth system.

Also, it is worth mentioning that for anyone coming from a physics background it will be very strange to discuss inertia in a first order equation. In classical mechanics, when the order of newton’s equations is reduced from second to first order, this is precisely the inertia-less limit. It is never made clear why this should be different here.

The primary issue, and I believe this has been climate textbook material for decades, is that different aspects of the climate have different timescales (i.e. the ocean is often thought to have much longer time scales) associated with them. The assumption of the present model is that a single bifurcation parameter and a forcing based on the mean state can accurately represent the system dynamics and in particular, the inertia in it. However, for the actual Earth System, it is the ocean that can provide inertia for changes and in the end I am unconvinced that the model presented has enough realism to provide much utility.

Finally, no information is given on how model complexity can be built up. In the AMOC box modeling literature there are one equation models, two equation models, three equation models, models that include switches (and hence bring in impulsive differential equation theory) and many more that I am likely overlooking.

I think this manuscript likely belongs in a mathematics journal where “inspired by applications” is good enough to motivate the mathematical analysis. I can imagine a version appearing in NPG (and I commend the authors on a clear writing style) but it would take a considerable amount of reworking.

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ED: Reconsider after major revisions (further review by editor and referees) (07 Jul 2026) by Vicente Perez-Munuzuri

AR by Chris Huntingford on behalf of the Authors (21 Jul 2026) Author's response Author's tracked changes Manuscript

ED: Publish as is (21 Jul 2026) by Vicente Perez-Munuzuri

AR by Chris Huntingford on behalf of the Authors (22 Jul 2026) Manuscript

Short summary

The risk of climate change triggering tipping points is a concern. While Earth System Models (ESMs) predict these points, their timing often varies. Equations of nonlinear dynamical systems can model tipping, hysteresis, and inertia, enabling comparison with ESMs. We propose a method to map tipping timing estimates onto a dynamical system that simulates a full hysteresis loop. We focus on solutions to the equation under overshoot forcing in global temperature and different inertia levels.