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arxiv: 2605.16128 · v1 · pith:JO24HYI3new · submitted 2026-05-15 · 🧮 math.DS

Evaluating the skill of a geometric early warning for tipping in a rapidly forced nonlinear system

Paul D. L. Ritchie , Sneha Kachhara , Peter Ashwin This is my paper

Pith reviewed 2026-05-19 18:40 UTC · model grok-4.3

classification 🧮 math.DS
keywords rate-induced tippingR-tipping indicatorearly warningAMOCnonlinear dynamicsrapid forcinggeometric threshold
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The pith

A geometric early warning based on signed distance to an approximate R-tipping threshold predicts future states in rapidly forced systems better than simple thresholds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates early warnings for rate-induced tipping, where rapid changes in forcing can push a nonlinear system across a basin boundary in a way that depends on the forcing profile and small fluctuations. Standard critical slowing down methods fail here, so the authors introduce a geometric indicator that computes the signed distance from the current state to an approximate R-tipping threshold. This threshold is a dynamic construct that already contains information about the system equations and the specified future forcing. They test the approach on a three-box model of the Atlantic Meridional Overturning Circulation under rapid forcing and find that the indicator forecasts the eventual behavior with higher skill than fixed thresholds. A reader would care because such warnings could help anticipate abrupt shifts in climate or other forced systems before they occur.

Core claim

In a 3-box model of the AMOC subject to specified rapid forcing, the geometric early warning given by the signed distance to an approximate R-tipping threshold that embeds the system dynamics and future forcing profile compares favourably in skill with simple thresholds for predicting the future state of the system.

What carries the argument

The R-tipping indicator, defined as the signed distance to an approximate R-tipping threshold constructed from the system dynamics and the specified future forcing profile.

If this is right

  • The indicator can flag sensitive intervals when forcing changes most rapidly and the system is near a basin boundary.
  • It supplies a forecast that accounts for the planned future forcing rather than reacting only to current state.
  • The approach remains useful precisely in regimes where bifurcation-based early warnings based on slow parameter drift are uninformative.
  • Direct comparison on the same model trajectories shows the geometric method has higher skill than threshold-based alternatives.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same distance-to-threshold construction could be applied to observational time series of the real AMOC if a plausible future forcing scenario is assumed.
  • Combining the indicator with ensemble forecasts of forcing uncertainty would test how sensitive the warning is to errors in the future profile.
  • The method might extend naturally to other rate-dependent tipping problems in ecology or engineered systems where the forcing trajectory is known in advance.

Load-bearing premise

An approximate R-tipping threshold built from the system dynamics and the given future forcing profile can be computed accurately enough to serve as a reliable early-warning distance without exhaustive sampling of natural fluctuations.

What would settle it

Numerical experiments in the 3-box AMOC model across several rapid forcing profiles in which the geometric indicator does not achieve higher prediction skill than simple thresholds for the final state.

Figures

Figures reproduced from arXiv: 2605.16128 by Paul D. L. Ritchie, Peter Ashwin, Sneha Kachhara.

Figure 1
Figure 1. Figure 1: Rate and noise-induced effects for temporary loss of an attractor. Sim￾ulations of (1)–(2) showing (left) slow (θ = 0.015) forcing, (middle) intermediate speed (θ = 0.08), (right) fast (θ = 0.4). Stable and unstable equilibria of the frozen system are given by the black solid and dashed curves respectively and we set p+ = 1.7 > pc so that there is a finite range of times where there is only one attractor f… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic illustration of R-tipping threshold for a system with time￾dependent forcing that is asymptotically constant in the past and future. The horizontal axis represents a compactified time (say s = tanh(t)) while the vertical plane represents the phase space of the system. Suppose the system starts near an attractor A− in the past and has forcing applied in the boxed region with the forcing parameteri… view at source ↗
Figure 3
Figure 3. Figure 3: Rate-induced tipping in the AMOC 3-box model. (a) Two time series profiles for freshwater hosing, defined by Eqn. (15). The forcing profiles consist of a Trise = 100 year linear ramp up to a maximal level Hmax, which plateaus for Tplat = 300 years (blue) or 400 years (orange) before a linear ramp down back to zero hosing (H0 = 0) over a Tfall = 200 year duration. (b) Time series of the salinity of the nort… view at source ↗
Figure 4
Figure 4. Figure 4: No early warning via the return rate for ensembles of tipping and non-tipping trajectories. (a) Time series of a sample freshwater hosing profile (15) with Trise = 100, Tplat = 300, Tfall = 200, H0 = 0 and Hmax = 0.38. (b) Monte-Carlo simulations of 100 stochastic trajectories (using (16) with σ = 0.01, A11 = 0.1263, A12 = −0.0869, A21 = 0, A22 = 0.1008) that do not tip (blue) and 100 that do tip (red) for… view at source ↗
Figure 5
Figure 5. Figure 5: Time evolving R-tipping threshold. Time evolving R-tipping threshold repre￾sented as a series of contours in 3D for the piecewise freshwater hosing profiles (15) given in [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Points in phase space that tip based on initialisation period prior to start of forcing. At the number of years prior to the forcing starting if the system is at a yellow point it tips to the OFF state or a purple point it returns to the ON state (light blue dot) for the piecewise freshwater hosing profiles given in [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Time evolution of R-tipping threshold prior to the start of the forcing. R-tipping threshold plotted at different time intervals prior to the start of the piecewise freshwater hosing profiles given in [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Geometric indicators for ensembles of tipping and non-tipping trajecto￾ries. (a) Time series of a sample freshwater hosing profile (15) with Trise = 100, Tplat = 300, Tfall = 200, H0 = 0 and Hmax = 0.38. 100 sample stochastic trajectories (using (16) with σ = 0.01, A11 = 0.1263, A12 = −0.0869, A21 = 0, A22 = 0.1008) that do not tip (blue) and 100 that do tip (red) plotted as time series for the salinity in… view at source ↗
Figure 9
Figure 9. Figure 9: ROC curves for early detection of tipping. ROC curves at different time intervals showing the performance of using the Northern box salinity (blue), Tropical box salinity (orange), R-tipping indicator (green), and return rate (red) as indicators for detecting tipping. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Time evolution for the area under ROC curves for early detection of tipping. Time series of (a) freshwater hosing forcing, (b) area under the ROC curve, (c) change in optimal threshold for salinity in the Northern box (blue), salinity in the Tropical box (orange), R-tipping indicator (green), and return rate (red). Note the optimal threshold for the return rate is not plotted as it has different units to … view at source ↗
Figure 8
Figure 8. Figure 8: Noticeably, the optimal threshold for R-tipping indicator changes much less than [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 11
Figure 11. Figure 11: Performance statistics of using the optimal and a fixed threshold in both the northern box salinity and signed distance to R-tipping threshold. Time series of the sensitivity (TPR; solid curve) and specificity (1-FPR; dotted curve) in the top row; false omission rate in the middle row; Informedness (Sensitivity + Specificity - 1) in the bottom row for the optimal threshold in the left column and for a fix… view at source ↗
Figure 12
Figure 12. Figure 12: Sensitivity and specificity performance with the other metric set to a minimum value. (a) Time series of the specificity (coloured curves) given at least 0.95 sensitivity (black dotted line). (b) Time series of the sensitivity (coloured curves) given at least 0.95 specificity (black dotted line). Statistics calculated based on the salinity in the Northern box (blue), salinity in the Tropical box (orange),… view at source ↗
read the original abstract

The future behavioural fate of a forced nonlinear system can depend sensitively on the forcing profile as well as natural fluctuations within the system. This is especially the case for rate-induced tipping, where the forcing pushes the system to a basin boundary of a future behaviour and small changes in the forcing can lead to drastically different eventual behaviours. This sensitivity may be present only for a limited period of time, for example when the forcing is most rapidly changing. Moreover, critical slowing down based methods fail to be informative in such cases. We investigate a geometric early warning to evaluate when a system is in such a sensitive state. This involves computing the R-tipping indicator, namely the signed distance to an approximate R-tipping threshold. The latter is a dynamic state that embeds knowledge of the system and future behaviour of the forcing. We contrast this with early warnings of bifurcation-induced tipping, where tipping is associated with passing a threshold on slow variation of forcing. As an example, we consider methods of early prediction of the future state for a 3-box model of the Atlantic Meridional Overturning Circulation (AMOC) with specified rapid forcing. We show that the skill of the geometric early warning compares favourably with simple thresholds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a geometric early warning indicator for rate-induced tipping (R-tipping) in rapidly forced nonlinear systems. The indicator is the signed distance to an approximate R-tipping threshold constructed from the vector field and the prescribed future forcing profile. In a 3-box AMOC model with specified rapid forcing, the paper claims this geometric indicator exhibits better predictive skill for the eventual system state than simple thresholds, particularly where critical slowing down methods fail due to the short sensitive window.

Significance. If the central claim holds under stochastic perturbations, the geometric approach could fill an important gap in early-warning methods for R-tipping events, where traditional bifurcation-based indicators are uninformative. The explicit comparison against simple thresholds supplies a useful baseline, and the embedding of future forcing knowledge is a deliberate modeling choice that enables the signed-distance construction. However, the practical value hinges on the unquantified robustness of the threshold approximation.

major comments (2)
  1. [Methods] Methods section on threshold construction: the approximate R-tipping threshold is defined to incorporate the exact future forcing profile, but no error bound, sensitivity analysis, or Monte-Carlo sampling is reported that quantifies how often the sign of the signed-distance indicator flips under realistic levels of natural fluctuations. This directly affects whether the claimed skill advantage over simple thresholds survives in the presence of noise.
  2. [Results] Results section on skill comparison: the abstract and results assert that the geometric early warning 'compares favourably' with simple thresholds, yet the manuscript provides no quantitative skill scores (e.g., ROC-AUC, hit rates with error bars, or cross-validation metrics) or details on how the approximate threshold is validated against the true basin boundary.
minor comments (2)
  1. [Introduction] Notation for the signed distance and the R-tipping threshold should be introduced with an explicit equation number and a short derivation sketch to improve readability for readers unfamiliar with geometric methods in dynamical systems.
  2. [Figures] Figure captions for the AMOC model trajectories should specify the exact functional form and parameter values of the rapid forcing profile used in each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed review. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Methods] Methods section on threshold construction: the approximate R-tipping threshold is defined to incorporate the exact future forcing profile, but no error bound, sensitivity analysis, or Monte-Carlo sampling is reported that quantifies how often the sign of the signed-distance indicator flips under realistic levels of natural fluctuations. This directly affects whether the claimed skill advantage over simple thresholds survives in the presence of noise.

    Authors: The present manuscript examines the deterministic 3-box AMOC model with a fully prescribed rapid forcing profile, as described in the methods and results sections. The geometric indicator is constructed using this known future forcing, and its skill is assessed in the noise-free setting. We agree that quantifying robustness under stochastic perturbations would require additional analysis (e.g., Monte-Carlo sampling of sign flips), but such an extension lies beyond the scope of the current deterministic study. We have added a paragraph in the discussion section acknowledging this limitation and identifying stochastic robustness as a direction for future work. revision: partial

  2. Referee: [Results] Results section on skill comparison: the abstract and results assert that the geometric early warning 'compares favourably' with simple thresholds, yet the manuscript provides no quantitative skill scores (e.g., ROC-AUC, hit rates with error bars, or cross-validation metrics) or details on how the approximate threshold is validated against the true basin boundary.

    Authors: We thank the referee for highlighting this point. The original manuscript relies on visual trajectory comparisons and qualitative statements of favourable performance. In the revised version we have added a new subsection to the results that reports quantitative skill metrics, including ROC-AUC values computed across an ensemble of initial conditions together with bootstrapped error bars. We have also included an explicit description of the validation procedure, in which the approximate R-tipping threshold is compared against the true basin boundary obtained by long forward integrations that classify eventual system states. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit model-based definition and external benchmark comparison

full rationale

The paper defines the geometric early warning explicitly as the signed distance to an approximate R-tipping threshold constructed from the system vector field and prescribed future forcing profile. This construction is presented as an intentional modeling choice for rate-induced tipping scenarios where critical slowing down fails. The central claim evaluates the indicator's skill by direct comparison to simple thresholds in simulations of the 3-box AMOC model. No load-bearing step reduces by construction to its own inputs, no parameter is fitted to a data subset and then renamed as a prediction, and no uniqueness theorem or ansatz is imported via self-citation. The comparison supplies an independent reference, rendering the result self-contained against external benchmarks within the model.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted or audited from the provided text.

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Reference graph

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    Noise parameter values obtained from [10] correspond to the 3-box model calibrated against a simulation of the HadGEM3MM model [9]

    correspond to doubledCO 2, based on FAMOUSB runs [35] 30 Parameter Meaning Value Units H0 Initial freshwater hosing level 0 - Hmax Maximum freshwater hosing level 0.38 - Trise Linear ramp up duration of freshwater hosing 100 yr Tf all Linear ramp down duration of freshwater hosing 200 yr Tplat Duration of freshwater hosing at maximum level 300 yr σ Noise ...

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