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arxiv: 2604.09661 · v1 · submitted 2026-03-30 · ⚛️ physics.ao-ph · nlin.AO· physics.data-an· stat.ME

Recognition: 2 theorem links

· Lean Theorem

Multistability and intermingledness in complex high-dimensional data

George Datseris , Johannes Lohmann , Ois\'in Hamilton , Jacob Haqq-Misra
Authors on Pith no claims yet

Pith reviewed 2026-05-14 00:13 UTC · model grok-4.3

classification ⚛️ physics.ao-ph nlin.AOphysics.data-anstat.ME
keywords multistabilityhigh-dimensional datanonlinear dynamicsclimate simulationssteady statestipping elementsinterminglednessoptimization routine
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The pith

A workflow detects alternative steady states in high-dimensional simulation data and identifies which observables best distinguish them.

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

The paper presents a computational workflow that applies nonlinear dynamics methods to finite high-dimensional datasets from simulations to algorithmically identify whether alternative steady states exist. An optimization routine within the framework determines which observables in the data most effectively differentiate those states from one another and which provide no distinction at all. This is demonstrated on three climate-related datasets involving Atlantic ocean circulation, midlatitude atmospheric flow, and exoplanet habitability. The work defines an intermingledness indicator that measures similarities and differences between the identified states and their basins of attraction across diagnostic variables.

Core claim

The central claim is that recent advances in computational nonlinear dynamics enable a workflow capable of analyzing potentially multistable high-dimensional simulation output to decide algorithmically what alternative steady states are present, if any. The framework runs an optimization that highlights the observables which best separate the states and those which do not differentiate them at all. Once states are located, the intermingledness measure quantifies how the states and their attraction basins relate to one another across chosen variables. The method is applied to three diverse climate datasets and supplied with open-source code for reuse.

What carries the argument

The multistability-analysis workflow that combines nonlinear-dynamics state identification with an optimization routine selecting differentiating observables, together with the intermingledness indicator for comparing states and basins.

If this is right

  • The optimization step can directly inform which variables to monitor for early warning of transitions in multistable climate or ecosystem components.
  • Once states are identified, the intermingledness measure supplies a quantitative comparison of how similar or distinct the states and their basins are across any chosen diagnostic variables.
  • The workflow applies to any finite high-dimensional simulation output suspected of multistability, not only the three climate examples shown.
  • Open-source implementation allows the same procedure to be run on new datasets without requiring manual state labeling in advance.

Where Pith is reading between the lines

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

  • If the optimization reliably flags non-differentiating observables, it could reduce the number of variables that need to be tracked in large ensemble simulations, lowering storage and analysis costs.
  • The same workflow might be tested on non-climate multistable systems such as neural-network training dynamics or ecological food-web models to see whether intermingledness reveals shared structural features.
  • A direct comparison between the workflow's detected basins and those obtained from long-term ensemble integrations on the same data would test how well finite simulation runs capture full basin geometry.

Load-bearing premise

Finite high-dimensional simulation outputs contain detectable multistable components whose alternative steady states can be reliably found and separated by the proposed workflow.

What would settle it

Apply the workflow to a controlled high-dimensional dataset whose multistable states have already been established by independent bifurcation analysis or ensemble runs, then check whether the optimization routine recovers the known differentiating observables and the intermingledness values match direct basin measurements.

Figures

Figures reproduced from arXiv: 2604.09661 by George Datseris, Jacob Haqq-Misra, Johannes Lohmann, Ois\'in Hamilton.

Figure 1
Figure 1. Figure 1: Summary of the steps followed by the workflow proposed in this paper, using as an [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Analysis of Veros data. The abbreviations of the diagnostic variables (’sst [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: As in Fig [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Continuation, as defined in §2.7, for the MOAOOAM dataset. Top panel: the centroids for each feature group (attractor) for varying the atmospheric emissivity ε, projected to the di￾agnostic variable psi a 3. This atmospheric mode corresponds to the barotropic streamfunction projected on an east-west gyre mode. The result is that larger or smaller centroids on this variable correspond to the a larger or wea… view at source ↗
Figure 5
Figure 5. Figure 5: Analysis of data produced by the ExoPlaSim model for the habitability of an exoplanet [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Reported silhouette mean s (quantity optimized to yield optimal DBSCAN clustering) during DBSCAN clustering for the Veros data of §3.1, versus number of groups created in the clustering. The marker shape corresponds to the number of features used, and markers are plotted with transparency. ocean or atmospheric flows which can then be used to understand climate characteristics of the model [Van+15; Ham25]. … view at source ↗
read the original abstract

Multistability is a phenomenon prevalent in many natural systems. In climate, for example, it allows the possibility of irreversible consequences on planetary scale as a result of climate change. Indeed, a climate ``tipping element'' is a multistable component that can undergo a transition to an alternative steady state due to an external perturbation. Despite the potential impact, multistability in realistic, complex simulations (e.g. climate models) remains poorly understood. Arguably a reason for this the lack of applicable methodology that explicitly targets finite yet high-dimensional datasets. In this work we utilize recent progress in computational nonlinear dynamics to formulate a workflow that analyses potentially multistable simulation data and decides algorithmically what are the alternative steady states contained within, if any. The framework undergoes an optimization routine that showcases which observables in the data best differentiate the alternative states, and which ones do not differentiate at all, which could be used to guide monitoring and early-warning for multistable components in climate or ecosystems. Finally, once the alternate states have been found, we define an indicator called ``intermingledness''. It quantifies differences and similarities between alternate states, as well as for their basins of attraction, across various diagnostic variables. We analyse and present results using three diverse climate datasets: Atlantic ocean circulation, atmospheric midlatitude flow, and habitability of exoplanets. We also provide easy-to-use open source code for applying the workflow to new data.

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 presents a workflow that combines tools from computational nonlinear dynamics with an optimization step to algorithmically identify alternative steady states in finite high-dimensional simulation output, select observables that best differentiate those states, and introduce a new 'intermingledness' metric that quantifies similarities and differences across states and their basins of attraction. The approach is demonstrated on three climate-related datasets (Atlantic meridional overturning circulation, midlatitude atmospheric flow, and exoplanet habitability) and accompanied by open-source code.

Significance. If the workflow proves robust, it would supply a practical, data-driven method for detecting multistability and tipping elements inside existing high-dimensional climate and ecosystem models, potentially guiding the choice of monitoring variables for early-warning systems. The open-source implementation is a clear strength that supports reproducibility and further testing.

major comments (2)
  1. [Methods] Methods section: the optimization routine used to rank differentiating observables is described at a high level but lacks explicit statements on convergence criteria, safeguards against local minima, and quantitative assessment of false-positive rates in state detection; these details are load-bearing for the central claim that the procedure reliably identifies multistable components.
  2. [Results] Results section (climate examples): no validation is performed on synthetic or analytically known multistable systems with controlled noise levels and known basin structures, so the reliability of the intermingledness indicator and the state-detection step cannot be assessed against ground truth.
minor comments (2)
  1. [Abstract/Introduction] The abstract and introduction use the term 'intermingledness' before it is formally defined; a brief forward reference or early definition would improve readability.
  2. [Figures] Figure captions should explicitly state the number of ensemble members or simulation length used for each dataset so that readers can judge statistical robustness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and for recognizing the potential utility of the workflow for climate and ecosystem applications. We address each major comment below, indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Methods] Methods section: the optimization routine used to rank differentiating observables is described at a high level but lacks explicit statements on convergence criteria, safeguards against local minima, and quantitative assessment of false-positive rates in state detection; these details are load-bearing for the central claim that the procedure reliably identifies multistable components.

    Authors: We agree that the optimization details require expansion to support the reliability of state identification. In the revised Methods section we will add: explicit convergence criteria based on a threshold on the change in the objective function over successive iterations; safeguards against local minima via multiple random initializations with selection of the lowest-cost solution; and a quantitative false-positive assessment using bootstrap resampling of the input trajectories to estimate variability in detected states. These additions will be included without changing the overall workflow or results. revision: yes

  2. Referee: [Results] Results section (climate examples): no validation is performed on synthetic or analytically known multistable systems with controlled noise levels and known basin structures, so the reliability of the intermingledness indicator and the state-detection step cannot be assessed against ground truth.

    Authors: We acknowledge that direct validation against synthetic systems with known ground truth would strengthen confidence in the intermingledness indicator and state detection. The manuscript prioritizes application to real high-dimensional climate data where ground truth is unavailable; however, we will add a short validation subsection in the revised Results that applies the workflow to a low-dimensional analytically tractable multistable system (e.g., a double-well potential with controlled additive noise) to demonstrate recovery of known basins. A comprehensive high-dimensional synthetic benchmark matching the complexity of the climate models lies outside the present scope and will be noted as a limitation for future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper describes an algorithmic workflow that first applies nonlinear-dynamics tools to identify alternative steady states in high-dimensional simulation output, then runs an optimization routine over observables and finally defines the intermingledness metric on the already-identified states and basins. No equation or step reduces by construction to a fitted parameter or to a self-citation whose content is the target result; the procedure is presented as a sequence of independent computational steps whose validity rests on the supplied open-source implementation rather than on any internal redefinition or uniqueness theorem imported from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only review prevents exhaustive enumeration; the workflow implicitly assumes multistable structure exists in the data and that nonlinear-dynamics tools can extract it, but no explicit free parameters, axioms, or invented entities beyond the new metric are stated.

axioms (1)
  • domain assumption Finite high-dimensional simulation data contains detectable multistable components identifiable by nonlinear dynamics methods
    Central premise required for the workflow to locate alternative steady states.
invented entities (1)
  • intermingledness indicator no independent evidence
    purpose: Quantifies differences and similarities between alternate states and their basins of attraction across diagnostic variables
    Newly defined metric introduced after state detection; no independent evidence outside the paper is provided.

pith-pipeline@v0.9.0 · 5576 in / 1292 out tokens · 68355 ms · 2026-05-14T00:13:45.344397+00:00 · methodology

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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