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arxiv: 2604.12869 · v1 · submitted 2026-04-14 · 🌊 nlin.CD · math.DS· physics.flu-dyn

Recognition: unknown

Precursors of extreme events and critical transitions

Riccardo Consonni , Luca Magri
Authors on Pith no claims yet

Pith reviewed 2026-05-10 13:50 UTC · model grok-4.3

classification 🌊 nlin.CD math.DSphysics.flu-dyn
keywords extreme eventscritical transitionscovariant Lyapunov vectorsfast-slow systemsdynamical systemsprecursorsnonlinear dynamicsprediction
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The pith

A cascade of stability changes in covariant Lyapunov vectors precedes extreme events, from which two precursors are derived that achieve perfect prediction.

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

The paper develops a dynamical systems explanation for extreme events and critical transitions in fast-slow nonlinear systems. It describes a specific sequence of three regimes that occur before such events: a slow regime in which fast covariant Lyapunov vectors remain tangent to fast eigenvectors and transversal to the slow subspace, a transition regime in which fast eigenvalues become neutrally stable and the vectors lose their tangency, and a critical regime in which a spectral gap forces the vectors to align along the dominant fast direction and breaks the transversality between subspaces. Two precursors are constructed directly from this cascade. Numerical tests on both low- and higher-dimensional examples show that the precursors detect every event and produce no false alarms.

Core claim

In fast-slow nonlinear systems, extreme events are preceded by a cascade of regimes identified through the covariant Lyapunov vectors and their eigenvalues. The sequence begins with a slow regime of tangent fast vectors that stay transversal to the slow subspace, continues through a transition regime of neutral stability where tangency is lost, and culminates in a critical regime where a strong spectral gap causes alignment along the dominant direction and breaks transversality. Two precursors built from observations of this cascade predict both extreme events and critical transitions with 100% precision and recall.

What carries the argument

The three-stage cascade of slow, transition, and critical regimes tracked via the tangency and transversality properties of covariant Lyapunov vectors and the stability of their eigenvalues.

If this is right

  • The two precursors provide time-forecasting capability for extreme events in the tested systems.
  • Critical transitions, treated as a subset of extreme events, are also predicted with the same perfect accuracy.
  • The cascade-based approach applies equally to low-dimensional and higher-dimensional examples.
  • Numerical evidence shows zero false positives or missed events under the tested conditions.

Where Pith is reading between the lines

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

  • If the cascade holds more generally, the precursors could be adapted to observational time series in applications such as fluid flows or climate variables.
  • Systems lacking a clear fast-slow separation might still benefit if analogous stability indicators can be defined.
  • Direct comparison of the precursors against existing early-warning signals on the same benchmark systems would clarify relative strengths.

Load-bearing premise

That this specific cascade of regimes in the covariant Lyapunov vectors always occurs before extreme events in fast-slow nonlinear systems.

What would settle it

An extreme event in a fast-slow nonlinear system that occurs without the preceding slow regime of transversal tangent vectors, neutral-stability transition, and critical regime of broken transversality.

Figures

Figures reproduced from arXiv: 2604.12869 by Luca Magri, Riccardo Consonni.

Figure 1
Figure 1. Figure 1: FIG. 1. Roadmap of the theoretical steps of Section III, Sec [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. On the left: CLV phase space structure when the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic of the pseudo-algorithm for angle-based [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Top: real part of the spectrum of the Jacobian in [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Bistable Rössler system chaotic attractor. [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Van der Pol system, CLVs of the fast-slow Van der [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Bistable Rössler system. Top: time evolution of the [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. FitzHugh–Nagumo coupled oscillator system, time [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. FitzHugh–Nagumo oscillator. First panel from [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Multiscale Lorenz 96 system. Top: time series of the [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Forewarning time of the precursors of extreme [PITH_FULL_IMAGE:figures/full_fig_p014_13.png] view at source ↗
read the original abstract

We propose a theory based on dynamical systems to explain and predict the occurrence of extreme events, of which critical transitions form a subset. In fast-slow nonlinear systems, we identify a cascade of events preceding extreme events: (i) a slow regime, in which the fast covariant Lyapunov vectors (CLVs) are both tangent to the fast eigenvectors and remain transversal to the slow subspace; (ii) a transition regime, in which the fast eigenvalues become neutrally stable while the fast CLVs are no longer tangent to the fast eigenvectors; and (iii) a critical regime, in which a strong spectral gap in the eigenvalues causes both fast and slow CLVs to become tangent along the dominant fast direction, breaking the transversality between fast and slow subspaces. Building on this cascade, we propose two precursors to forewarn the occurrence of extreme events. We numerically test the theory and precursors on low- and higher-dimensional systems. The proposed precursors predict extreme events and critical transitions with 100% precision and recall. This work opens opportunities for time-forecasting extreme events using theoretically grounded precursors.

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 dynamical systems theory for extreme events (including critical transitions) in fast-slow nonlinear systems. It identifies a three-regime cascade preceding such events based on covariant Lyapunov vectors (CLVs) and eigenvalue behavior: (i) slow regime with fast CLVs tangent to fast eigenvectors yet transversal to the slow subspace; (ii) transition regime with neutral fast eigenvalues; (iii) critical regime with spectral-gap-induced tangency that breaks transversality. Two precursors are derived from this cascade and tested numerically on low- and higher-dimensional systems, with the claim that they achieve 100% precision and recall.

Significance. If the cascade proves general and the precursors robust, the work would provide a valuable theoretically grounded framework for early warning of extreme events, building on established tools like CLVs. The reported perfect performance on the tested systems is a clear strength that merits credit, as is the explicit linkage of precursors to the identified dynamical regimes rather than purely data-driven fitting.

major comments (2)
  1. [Abstract] Abstract: The central claim that the precursors achieve '100% precision and recall' is load-bearing for the paper's contribution, yet no information is provided on the specific systems tested (equations, dimensions, parameter values), number of extreme events, data exclusion rules, error bars, or whether the cascade and precursors were identified before or after inspecting the data. This prevents assessment of whether the perfect scores reflect genuine predictive power or post-hoc selection.
  2. [Numerical tests] Theory and numerical tests: The manuscript does not establish that the three-regime cascade is necessary for extreme events in fast-slow systems in general, nor does it include counter-example searches or tests under modest changes in time-scale separation or parameters. The evidence is confined to the chosen low- and higher-dimensional examples, so the universality required for the precursors to be general early-warning signals remains unproven.
minor comments (2)
  1. [Theory] The definition and computation of the fast and slow CLVs, including how transversality is quantified, should be stated explicitly with equations rather than assumed from prior literature.
  2. [Numerical tests] A brief discussion of how the precursors behave when the time-scale separation is varied continuously (rather than fixed) would strengthen the presentation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and positive assessment of the work's potential significance. We address each major comment below with specific plans for revision where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that the precursors achieve '100% precision and recall' is load-bearing for the paper's contribution, yet no information is provided on the specific systems tested (equations, dimensions, parameter values), number of extreme events, data exclusion rules, error bars, or whether the cascade and precursors were identified before or after inspecting the data. This prevents assessment of whether the perfect scores reflect genuine predictive power or post-hoc selection.

    Authors: The full details on the tested systems (including governing equations, dimensions, and parameter values), the number of extreme events analyzed across multiple realizations, data exclusion criteria, and statistical reporting are contained in the Numerical Tests section. The cascade was derived from the covariant Lyapunov vector and eigenvalue analysis prior to any numerical validation, and the precursors follow directly from the three-regime structure rather than data-driven fitting. To make this transparent in the abstract itself, we will add a concise clause summarizing the low- and higher-dimensional examples and the scale of the tests performed. We will also ensure error bars from ensemble runs are explicitly reported in the figures and text. revision: yes

  2. Referee: [Numerical tests] Theory and numerical tests: The manuscript does not establish that the three-regime cascade is necessary for extreme events in fast-slow systems in general, nor does it include counter-example searches or tests under modest changes in time-scale separation or parameters. The evidence is confined to the chosen low- and higher-dimensional examples, so the universality required for the precursors to be general early-warning signals remains unproven.

    Authors: We do not claim to have proven that the cascade is necessary in every fast-slow system; the manuscript presents a dynamical mechanism observed and analyzed in representative cases, together with precursors derived from it. In the revised version we will add an explicit limitations paragraph stating the current scope, perform additional numerical checks under modest variations of the time-scale separation parameter, and include a brief search for counter-examples in systems lacking a clear spectral gap. These steps will better delineate the conditions under which the precursors are expected to apply, while acknowledging that a fully general proof lies beyond the present scope. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper's central chain proceeds from standard dynamical-systems concepts (covariant Lyapunov vectors, spectral gaps, transversality between fast/slow subspaces) to an observed cascade of regimes, then to two derived precursors whose definitions are stated explicitly in terms of those CLV and eigenvalue properties. Numerical tests on chosen low- and higher-dimensional examples are presented as validation rather than as the source of the definitions themselves; no equation or claim reduces a fitted parameter or self-citation back into the reported 100 % precision/recall figures by construction. The absence of load-bearing self-citations, ansatz smuggling, or renaming of known empirical patterns keeps the derivation independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and universality of the described CLV/eigenvalue cascade in fast-slow systems; no explicit free parameters, new entities, or ad-hoc axioms are stated in the abstract.

axioms (1)
  • domain assumption Fast-slow nonlinear systems exhibit the three-regime cascade in covariant Lyapunov vectors and eigenvalues as described.
    This is the load-bearing premise extracted from the abstract; its validity is not demonstrated within the provided text.

pith-pipeline@v0.9.0 · 5488 in / 1384 out tokens · 51535 ms · 2026-05-10T13:50:15.560729+00:00 · methodology

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