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arxiv: 2602.08022 · v2 · submitted 2026-02-08 · ❄️ cond-mat.stat-mech · nlin.CD· physics.ao-ph· physics.data-an

Linear Response and Optimal Fingerprinting for Nonautonomous Systems

Valerio Lucarini This is my paper

Pith reviewed 2026-05-16 06:02 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech nlin.CDphysics.ao-phphysics.data-an
keywords linear response theoryoptimal fingerprintingnonautonomous systemstime-dependent Markov chainsdiffusion processesclimate change attributionpullback measuresequivariant measure
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The pith

Linear response formulas for time-dependent Markov chains and diffusions extend optimal fingerprinting to non-stationary backgrounds.

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

The paper derives explicit formulas for linear response in systems whose transition rules change with time, including Markov chains and diffusion processes. This matters because it supplies the mathematical bridge needed to predict how external forcings alter an evolving reference state and to attribute observed shifts back to those forcings. A reader would care if the same tools already used for stationary climate attribution could be applied when the background itself is drifting, as occurs during ongoing global change. The authors demonstrate the formulas on a modified energy-balance model, confirming that response predictions remain accurate even in coarse-grained settings and that the extended fingerprinting method can separate signals from several simultaneous forcings across long time intervals.

Core claim

We provide a link between response theory, pullback measures, and optimal fingerprinting that allows predicting the impact of acting forcings on time-dependent systems and attributing observed anomalies when the reference state is not time-independent. Formulas are derived for linear response theory for time-dependent Markov chains and diffusion processes. Existence, uniqueness, and differentiability of the equivariant measure are established under general perturbations of the transition kernels. These results extend optimal fingerprinting for detection and attribution to time-dependent backgrounds and to the simultaneous treatment of multiple time slices, with numerical verification on a Gh

What carries the argument

The equivariant measure for time-dependent transition kernels, whose differentiability supplies the linear response formulas and connects them to pullback measures.

If this is right

  • Response predictions become available for any system whose background state evolves, not only stationary ones.
  • Optimal fingerprinting can now be solved jointly across many time slices rather than slice by slice.
  • Attribution of observed anomalies to multiple concurrent forcings becomes feasible in non-stationary regimes.
  • Coarse-grained Markov or diffusion models can still deliver accurate response estimates for practical applications.

Where Pith is reading between the lines

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

  • The same construction could be tested on real observational records to see whether attribution skill improves when non-stationarity is explicitly included.
  • If the differentiability property holds more broadly, it would open a route to second-order response corrections for stronger forcings.
  • The framework naturally suggests analogous extensions for other nonautonomous complex systems such as ecological networks or financial markets.
  • Numerical experiments with finer spatial resolution or stochastic noise terms would clarify the robustness of the coarse-grained accuracy reported here.

Load-bearing premise

The equivariant measure must exist, remain unique, and stay differentiable under arbitrary perturbations of the transition kernels.

What would settle it

A side-by-side comparison in the energy-balance model where the predicted temperature response to time-varying CO2 differs measurably from direct simulation results would show the formulas do not hold.

Figures

Figures reproduced from arXiv: 2602.08022 by Valerio Lucarini.

Figure 1
Figure 1. Figure 1: a) Ensemble average of simulations performed considering exclusively the sun spots cycle and volcanic eruptions as forcings to the system. b) Logarithm of the probability distribution of the yearly values of TAV E and ∆T for the reference time-dependent system. We have used 10000 ensemble members. The Voronoi tessellation used for constructing the reduced Markov chain is shown. The arrows provide a qualita… view at source ↗
Figure 2
Figure 2. Figure 2: Optimal fingerprinting for detection and attribution of climate change for a nonautonomous reference state. a) Evolution of the annual anomaly of the temperature field for one ensemble member undergoing both the CO2 and aerosols forcing. b) CO2 forcing fingerprint XCO2 . c) Aerosols forcing fingerprint XA. We now perform Nens = 10000 simulations where we include the natural forcing, the CO2 forcing, and th… view at source ↗
Figure 3
Figure 3. Figure 3: Optimal fingerprinting for detection and attribution of climate change for a nonautonomous reference state. a) Average value ± 2 standard deviations computed across the ensemble simulations of the weighting factor the CO2 (βCO2 ) and aerosols (βA) fingerprints. The corresponding temporal modulations of the forcing n(t) (cor CO2) and γ(t) (for aersolos) are shown in the insets. b) Scatter plot of the β fact… view at source ↗
read the original abstract

We provide a link between response theory, pullback measures, and optimal fingerprinting method that paves the way for a) predicting the impact of acting forcings on time-dependent systems and b) attributing observed anomalies to acting forcings when the reference state is not time-independent. We derive formulas for linear response theory for time-dependent Markov chains and diffusion processes. We discuss existence, uniqueness, and differentiability of the equivariant measure under general (not necessarily slow or periodic) perturbations of the transition kernels. Our results allow for extending the theory of optimal fingerprinting for detection and attribution of climate change (or change in any complex system) when the background state is time-dependent amd when the optimal solution is sought for multiple time slices at the same time. We provide numerical support for the findings by applying our theory to a modified version of the Ghil-Sellers energy balance model. We verify the precision of response theory - even in a coarse-grained setting - in predicting the impact of increasing CO$_2$ concentration on the temperature field. Additionally, we show that the optimal fingerprinting method developed here is capable to attribute the climate change signal to multiple acting forcings across a vast time horizon.

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 paper derives linear response formulas for nonautonomous Markov chains and diffusion processes via pullback/equivariant measures, establishes existence/uniqueness/differentiability of these measures under general (not necessarily slow or periodic) perturbations of the transition kernels, and extends optimal fingerprinting to time-dependent background states for simultaneous multi-slice detection and attribution. Numerical tests on a modified Ghil-Sellers energy-balance model are used to verify predictive accuracy for CO2 forcing and attribution to multiple forcings over long time horizons.

Significance. If the central derivations are rigorous and the required hypotheses on the kernels are made explicit, the work would meaningfully extend linear response and optimal fingerprinting from autonomous to nonautonomous settings, directly addressing a practical limitation in climate detection/attribution when the reference climate is itself evolving. The numerical validation on a coarse-grained climate model is a positive indicator of practical utility.

major comments (2)
  1. [Abstract / theoretical section on pullback measures] Abstract and theoretical development of equivariant measures: the claim that differentiability of the equivariant measure μ_t holds for 'general' (arbitrary) perturbations of the transition kernels P_t is stated without the necessary hypotheses (uniform-in-time spectral gap, uniform ergodicity, or C^1 regularity in both space and time). These conditions are load-bearing; if they fail at even one instant the derivative dμ_t/dε ceases to exist in the required sense and the subsequent optimal-fingerprinting formulas collapse. Explicit statement of the minimal assumptions (or a counter-example showing when they can be relaxed) is required.
  2. [Optimal fingerprinting extension] Extension to optimal fingerprinting: the multi-time-slice formulation is presented as a direct consequence of the linear-response formulas, yet the paper does not demonstrate that the resulting optimization problem remains well-posed (unique solution, stable inversion) once the time-dependent background measure is substituted. A concrete derivation or numerical check of the Hessian or conditioning of the fingerprinting operator under time-dependent μ_t is needed.
minor comments (2)
  1. [Abstract] Abstract contains a typographical error: 'amd' should read 'and'.
  2. [Throughout theoretical sections] Notation for the time-dependent kernels and measures should be introduced once and used consistently; occasional switches between P_t, K_t, μ_t and pullback notation make the derivations harder to follow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight important points for improving the rigor and clarity of the presentation. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract / theoretical section on pullback measures] Abstract and theoretical development of equivariant measures: the claim that differentiability of the equivariant measure μ_t holds for 'general' (arbitrary) perturbations of the transition kernels P_t is stated without the necessary hypotheses (uniform-in-time spectral gap, uniform ergodicity, or C^1 regularity in both space and time). These conditions are load-bearing; if they fail at even one instant the derivative dμ_t/dε ceases to exist in the required sense and the subsequent optimal-fingerprinting formulas collapse. Explicit statement of the minimal assumptions (or a counter-example showing when they can be relaxed) is required.

    Authors: We agree that the hypotheses underlying the differentiability of the equivariant measure must be stated explicitly. Our derivations rely on the existence of a uniform-in-time spectral gap for the family of transition kernels together with appropriate regularity (C^1 in space and time) to guarantee the required differentiability. In the revised manuscript we will add a dedicated paragraph (or subsection) that lists these minimal assumptions at the outset of the theoretical development, referencing standard results on nonautonomous Markov processes. We will also note that the results do not hold without such conditions and briefly indicate why they are necessary. revision: yes

  2. Referee: [Optimal fingerprinting extension] Extension to optimal fingerprinting: the multi-time-slice formulation is presented as a direct consequence of the linear-response formulas, yet the paper does not demonstrate that the resulting optimization problem remains well-posed (unique solution, stable inversion) once the time-dependent background measure is substituted. A concrete derivation or numerical check of the Hessian or conditioning of the fingerprinting operator under time-dependent μ_t is needed.

    Authors: We acknowledge the need to verify well-posedness of the multi-time-slice optimization under a time-dependent background. In the revision we will insert a short derivation showing that the Hessian of the fingerprinting objective remains positive definite when the background is replaced by the equivariant measure μ_t, using the linear-response representation of the response operator. We will also augment the numerical section with explicit condition-number diagnostics for the fingerprinting matrices computed on the Ghil-Sellers model across the multi-decadal time windows already considered, thereby confirming numerical stability of the inversion. revision: yes

Circularity Check

0 steps flagged

Derivations for linear response in nonautonomous systems are self-contained first-principles extensions

full rationale

The paper derives linear response formulas for time-dependent Markov chains and diffusions by extending standard response theory to pullback/equivariant measures, with explicit discussion of existence, uniqueness, and differentiability under general kernel perturbations. These steps are presented as mathematical constructions independent of the target fingerprinting application. Numerical verification on the modified Ghil-Sellers model provides an external check that does not reduce to any internal fit or self-citation chain. No load-bearing step equates a claimed prediction to a fitted parameter or prior self-citation by construction; the central claims remain independently verifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the mathematical properties of equivariant measures for nonautonomous systems; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Existence, uniqueness, and differentiability of the equivariant measure under general perturbations of the transition kernels
    Invoked to guarantee the linear response formulas and the extension of fingerprinting to time-dependent cases.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A Mathematical Framework for Linear Response Theory for Nonautonomous Systems

    math.DS 2026-03 unverdicted novelty 8.0

    Establishes rigorous linear response formulas for general deterministic and random nonautonomous systems with fast memory loss via a global transfer operator on sequence space of measures.

Reference graph

Works this paper leans on

133 extracted references · 133 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Kubo, Statistical-mechanical theory of irreversible processes

    Kubo R. 1957 Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems.J. Phys. Soc. Japan12, 570–586. (10.1143/JPSJ.12.570)

    work page doi:10.1143/jpsj.12.570 1957

  2. [2]

    1966 The fluctuation-dissipation theorem.Rep

    Kubo R. 1966 The fluctuation-dissipation theorem.Rep. Prog. Phys.29, 255–284

    work page 1966

  3. [3]

    2010 A simple framework to justify linear response theory.Nonlinearity 23, 909–922

    Hairer M, Majda AJ. 2010 A simple framework to justify linear response theory.Nonlinearity 23, 909–922. (10.1088/0951-7715/23/4/008)

    work page doi:10.1088/0951-7715/23/4/008 2010

  4. [4]

    2005 Off-equilibrium generalization of the fluctuation dissipation theorem for Ising spins and measurement of the linear response function.Phys

    Lippiello E, Corberi F, Zannetti M. 2005 Off-equilibrium generalization of the fluctuation dissipation theorem for Ising spins and measurement of the linear response function.Phys. Rev. E71, 036104. (10.1103/PhysRevE.71.036104)

    work page doi:10.1103/physreve.71.036104 2005

  5. [5]

    2008 Fluctuation-Dissipation: Response Theory in Statistical Physics.Phys

    Marconi UMB, Puglisi A, Rondoni L, Vulpiani A. 2008 Fluctuation-Dissipation: Response Theory in Statistical Physics.Phys. Rep.461, 111

    work page 2008

  6. [6]

    2009 Fluctuations and Response of Nonequilibrium States.Phys

    Baiesi M, Maes C, Wynants B. 2009 Fluctuations and Response of Nonequilibrium States.Phys. Rev. Lett.103, 010602. (10.1103/PhysRevLett.103.010602)

    work page doi:10.1103/physrevlett.103.010602 2009

  7. [7]

    2013 An update on the nonequilibrium linear response.New Journal of Physics15, 013004

    Baiesi M, Maes C. 2013 An update on the nonequilibrium linear response.New Journal of Physics15, 013004. (10.1088/1367-2630/15/1/013004) 23royalsocietypublishing.org/journal/rspa Proc R Soc A 0000000

    work page doi:10.1088/1367-2630/15/1/013004 2013

  8. [8]

    Thermal effects and their compen- sation in advanced virgo.Journal of Physics: Confer- ence Series, 363:012016, jun 2012

    Colangeli M, Lucarini V . 2014 Elements of a unified framework for response formulae. Journal of Statistical Mechanics: Theory and Experiment2014, P01002. (10.1088/1742- 5468/2014/01/p01002)

    work page doi:10.1088/1742- 2014

  9. [9]

    2019 On the fluctuation-dissipation relation in non-equilibrium and non-Hamiltonian systems.Chaos: An Interdisciplinary Journal of Nonlinear Science29, 083132

    Sarracino A, Vulpiani A. 2019 On the fluctuation-dissipation relation in non-equilibrium and non-Hamiltonian systems.Chaos: An Interdisciplinary Journal of Nonlinear Science29, 083132. (10.1063/1.5110262)

    work page doi:10.1063/1.5110262 2019

  10. [10]

    2019 Linear response for macroscopic observables in high- dimensional systems.Chaos: An Interdisciplinary Journal of Nonlinear Science29, 113127

    Wormell CL, Gottwald GA. 2019 Linear response for macroscopic observables in high- dimensional systems.Chaos: An Interdisciplinary Journal of Nonlinear Science29, 113127. (10.1063/1.5122740)

    work page doi:10.1063/1.5122740 2019

  11. [11]

    1998 Nonequilibrium statistical mechanics near equilibrium: computing higher- order terms.Nonlinearity11, 5–18

    Ruelle D. 1998 Nonequilibrium statistical mechanics near equilibrium: computing higher- order terms.Nonlinearity11, 5–18

    work page 1998

  12. [12]

    2005 Nonlinear susceptibility in glassy systems: A probe for cooperative dynamical length scales.Phys

    Bouchaud JP , Biroli G. 2005 Nonlinear susceptibility in glassy systems: A probe for cooperative dynamical length scales.Phys. Rev. B72, 064204. (10.1103/PhysRevB.72.064204)

    work page doi:10.1103/physrevb.72.064204 2005

  13. [13]

    2008 Nonlinear susceptibilities and the measurement of a cooperative length.Phys

    Lippiello E, Corberi F, Sarracino A, Zannetti M. 2008 Nonlinear susceptibilities and the measurement of a cooperative length.Phys. Rev. B77, 212201. (10.1103/PhysRevB.77.212201)

    work page doi:10.1103/physrevb.77.212201 2008

  14. [14]

    2008 Response Theory for Equilibrium and Non-Equilibrium Statistical Mechanics: Causality and Generalized Kramers-Kronig relations.Journal of Statistical Physics 131, 543–558

    Lucarini V . 2008 Response Theory for Equilibrium and Non-Equilibrium Statistical Mechanics: Causality and Generalized Kramers-Kronig relations.Journal of Statistical Physics 131, 543–558. (10.1007/s10955-008-9498-y)

    work page doi:10.1007/s10955-008-9498-y 2008

  15. [15]

    2012 Beyond the linear fluctuation-dissipation theorem: the role of causality.Journal of Statistical Mechanics: Theory and Experiment2012, P05013

    Lucarini V , Colangeli M. 2012 Beyond the linear fluctuation-dissipation theorem: the role of causality.Journal of Statistical Mechanics: Theory and Experiment2012, P05013

    work page 2012

  16. [16]

    2015 Frenetic aspects of second order response.Phys

    Basu U, Krüger M, Lazarescu A, Maes C. 2015 Frenetic aspects of second order response.Phys. Chem. Chem. Phys.17, 6653–6666. (10.1039/C4CP04977B)

    work page doi:10.1039/c4cp04977b 2015

  17. [17]

    Lucarini V , Wouters J. 2017 Response formulae for n-point correlations in statistical mechanical systems and application to a problem of coarse graining.Journal of Physics A: Mathematical and Theoretical50, 355003. (10.1088/1751-8121/aa812c)

    work page doi:10.1088/1751-8121/aa812c 2017

  18. [18]

    1998 General linear response formula in statistical mechanics, and the fluctuation- dissipation theorem far from equilibrium.Phys

    Ruelle D. 1998 General linear response formula in statistical mechanics, and the fluctuation- dissipation theorem far from equilibrium.Phys. Lett. A245, 220–224. (10.1016/S0375- 9601(98)00419-8)

    work page doi:10.1016/s0375- 1998

  19. [19]

    2009 A review of linear response theory for general differentiable dynamical systems.Nonlinearity22, 855–870

    Ruelle D. 2009 A review of linear response theory for general differentiable dynamical systems.Nonlinearity22, 855–870. (10.1088/0951-7715/22/4/009)

    work page doi:10.1088/0951-7715/22/4/009 2009

  20. [20]

    2007 Blended response algorithms for linear fluctuation-dissipation for complex nonlinear dynamical systems.Nonlinearity20, 2793

    Abramov R, Majda A. 2007 Blended response algorithms for linear fluctuation-dissipation for complex nonlinear dynamical systems.Nonlinearity20, 2793

    work page 2007

  21. [21]

    2021 Approximating Linear Response by Nonintrusive Shadowing Algorithms.SIAM Journal on Numerical Analysis59, 2843–2865

    Ni A. 2021 Approximating Linear Response by Nonintrusive Shadowing Algorithms.SIAM Journal on Numerical Analysis59, 2843–2865. (10.1137/20M1388255)

    work page doi:10.1137/20m1388255 2021

  22. [22]

    2022 Efficient Computation of Linear Response of Chaotic Attractors with One-Dimensional Unstable Manifolds.SIAM Journal on Applied Dynamical Systems21, 735–781

    Chandramoorthy N, Wang Q. 2022 Efficient Computation of Linear Response of Chaotic Attractors with One-Dimensional Unstable Manifolds.SIAM Journal on Applied Dynamical Systems21, 735–781. (10.1137/21M1405599)

    work page doi:10.1137/21m1405599 2022

  23. [23]

    2023 Fast Adjoint Algorithm for Linear Responses of Hyperbolic Chaos.SIAM Journal on Applied Dynamical Systems22, 2792–2824

    Ni A. 2023 Fast Adjoint Algorithm for Linear Responses of Hyperbolic Chaos.SIAM Journal on Applied Dynamical Systems22, 2792–2824. (10.1137/22M1522383)

    work page doi:10.1137/22m1522383 2023

  24. [24]

    Spectral

    Mezi´ c I. 2005 Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dynam.41, 309–325. (10.1007/s11071-005-2824-x)

    work page doi:10.1007/s11071-005-2824-x 2005

  25. [25]

    Budiˇ si´ c, R

    Budiši´ c M, Mohr R, Mezi´ c I. 2012 Applied Koopmanism.Chaos22, 047510, 33. (10.1063/1.4772195)

    work page doi:10.1063/1.4772195 2012

  26. [26]

    Modern koopman theory for dynamical systems,

    Brunton SL, Budiši´ c M, Kaiser E, Kutz JN. 2022 Modern Koopman Theory for Dynamical Systems.SIAM Review64, 229–340. (10.1137/21M1401243)

    work page doi:10.1137/21m1401243 2022

  27. [27]

    Maggiore, J

    Gutiérrez MS, Lucarini V . 2022 On some aspects of the response to stochastic and deterministic forcings.Journal of Physics A: Mathematical and Theoretical55, 425002. (10.1088/1751- 8121/ac90fd)

    work page doi:10.1088/1751- 2022

  28. [28]

    2023 Theoretical tools for understanding the climate crisis from Hasselmann’s programme and beyond.Nature Reviews Physics5, 744–765

    Lucarini V , Chekroun MD. 2023 Theoretical tools for understanding the climate crisis from Hasselmann’s programme and beyond.Nature Reviews Physics5, 744–765. (10.1038/s42254- 023-00650-8)

    work page doi:10.1038/s42254- 2023

  29. [29]

    A general framework for linking free and forced fluctuations via Koopmanism , journal =

    Lucarini V , Gutiérrez MS, Moroney J, Zagli N. 2026 A general framework for linking free and forced fluctuations via Koopmanism.Chaos, Solitons and Fractals202, 117540. (https://doi.org/10.1016/j.chaos.2025.117540)

    work page doi:10.1016/j.chaos.2025.117540 2026

  30. [30]

    2026 Bridging the gap between Koopmanism and response theory: Using natural variability to predict forced response.SIAM Journal on Applied Dynamical Systems25, 196–229

    Zagli N, Colbrook MJ, Lucarini V , Mezi´ c I, Moroney J. 2026 Bridging the gap between Koopmanism and response theory: Using natural variability to predict forced response.SIAM Journal on Applied Dynamical Systems25, 196–229

    work page 2026

  31. [31]

    2025 Kolmogorov modes and linear response of jump- diffusion models.Reports on Progress in Physics88, 127601

    Chekroun MD, Zagli N, Lucarini V . 2025 Kolmogorov modes and linear response of jump- diffusion models.Reports on Progress in Physics88, 127601. (10.1088/1361-6633/ae2206) 24royalsocietypublishing.org/journal/rspa Proc R Soc A 0000000

    work page doi:10.1088/1361-6633/ae2206 2025

  32. [32]

    2020 Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space

    Chekroun MD, Tantet A, Dijkstra HA, Neelin JD. 2020 Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part I: Theory.Journal of Statistical Physics. (10.1007/s10955-020-02535-x)

    work page doi:10.1007/s10955-020-02535-x 2020

  33. [33]

    2024 Detecting and Attributing Change in Climate and Complex Systems: Foundations, Green’s Functions, and Nonlinear Fingerprints.Phys

    Lucarini V , Chekroun MD. 2024 Detecting and Attributing Change in Climate and Complex Systems: Foundations, Green’s Functions, and Nonlinear Fingerprints.Phys. Rev. Lett.133, 244201. (10.1103/PhysRevLett.133.244201)

    work page doi:10.1103/physrevlett.133.244201 2024

  34. [34]

    Theory of dynamic critical phenomena

    Hohenberg PC, Halperin BI. 1977 Theory of dynamic critical phenomena.Rev. Mod. Phys.49, 435–479. (10.1103/RevModPhys.49.435)

    work page doi:10.1103/revmodphys.49.435 1977

  35. [35]

    2009 Early-warning signals for critical transitions.Nature461, 53–59

    Scheffer M, Bascompte J, Brock WA, Brovkin V , Carpenter SR, Dakos V , Held H, Van Nes EH, Rietkerk M, Sugihara G. 2009 Early-warning signals for critical transitions.Nature461, 53–59

    work page 2009

  36. [36]

    2012 Early warning of climate tipping points from critical slowing down: comparing methods to improve robustness

    Lenton TM, Livina VN, Dakos V , van Nes EH, Scheffer M. 2012 Early warning of climate tipping points from critical slowing down: comparing methods to improve robustness. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, 1185–1204. (10.1098/rsta.2011.0304)

    work page doi:10.1098/rsta.2011.0304 2012

  37. [37]

    2024 Tipping point detection and early warnings in climate, ecological, and human systems.Earth System Dynamics15, 1117–1135

    Dakos V , Boulton CA, Buxton JE, Abrams JF, Arellano-Nava B, Armstrong McKay DI, Bathiany S, Blaschke L, Boers N, Dylewsky D et al.. 2024 Tipping point detection and early warnings in climate, ecological, and human systems.Earth System Dynamics15, 1117–1135

    work page 2024

  38. [38]

    2024 Chapter 4 - The multiverse of dynamic mode decomposition algorithms

    Colbrook MJ. 2024 Chapter 4 - The multiverse of dynamic mode decomposition algorithms. In Mishra S, Townsend A, editors,Numerical Analysis Meets Machine Learning, Handbook of Numerical Analysis, vol. 25, pp. 127–230. Elsevier. (https://doi.org/10.1016/bs.hna.2024.05.004)

    work page doi:10.1016/bs.hna.2024.05.004 2024

  39. [39]

    2024 Beyond expectations: residual dynamic mode decomposition and variance for stochastic dynamical systems.Nonlinear Dynamics112, 2037–

    Colbrook MJ, Li Q, Raut RV , Townsend A. 2024 Beyond expectations: residual dynamic mode decomposition and variance for stochastic dynamical systems.Nonlinear Dynamics112, 2037–

    work page 2024

  40. [40]

    (10.1007/s11071-023-09135-w)

    work page doi:10.1007/s11071-023-09135-w

  41. [41]

    2024 Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems.Communications on Pure and Applied Mathematics 77, 221–283

    Colbrook MJ, Townsend A. 2024 Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems.Communications on Pure and Applied Mathematics 77, 221–283. (https://doi.org/10.1002/cpa.22125)

    work page doi:10.1002/cpa.22125 2024

  42. [42]

    2015 A kernel-based method for data- driven koopman spectral analysis.Journal of Computational Dynamics2, 247–265

    Williams MO, Rowley CW, Kevrekidis IG. 2015 A kernel-based method for data- driven koopman spectral analysis.Journal of Computational Dynamics2, 247–265. (10.3934/jcd.2015005)

    work page doi:10.3934/jcd.2015005 2015

  43. [43]

    2020 Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator.Entropy22

    Klus S, Nüske F, Hamzi B. 2020 Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator.Entropy22. (10.3390/e22070722)

    work page doi:10.3390/e22070722 2020

  44. [44]

    1991 Voronoi diagrams—a survey of a fundamental geometric data structure.ACM Comput

    Aurenhammer F. 1991 Voronoi diagrams—a survey of a fundamental geometric data structure.ACM Comput. Surv.23, 345–405. (10.1145/116873.116880)

    work page doi:10.1145/116873.116880 1991

  45. [45]

    1998Markov chains

    Norris J. 1998Markov chains. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge University Press

    work page

  46. [46]

    2010 Everything you wanted to know about Markov State Models but were afraid to ask.Methods52, 99–105

    Pande VS, Beauchamp K, Bowman GR. 2010 Everything you wanted to know about Markov State Models but were afraid to ask.Methods52, 99–105. (10.1016/j.ymeth.2010.06.002)

    work page doi:10.1016/j.ymeth.2010.06.002 2010

  47. [47]

    2012 Optimal control of molecular dynamics using Markov state models.Mathematical Programming134, 259–282

    Schuette C, Winkelmann S, Hartmann C. 2012 Optimal control of molecular dynamics using Markov state models.Mathematical Programming134, 259–282. (10.1007/s10107-012-0547-6)

    work page doi:10.1007/s10107-012-0547-6 2012

  48. [48]

    2018 Markov State Models: From an Art to a Science.Journal of the American Chemical Society140, 2386–2396

    Husic BE, Pande VS. 2018 Markov State Models: From an Art to a Science.Journal of the American Chemical Society140, 2386–2396. doi: 10.1021/jacs.7b12191 (10.1021/jacs.7b12191)

    work page doi:10.1021/jacs.7b12191 2018

  49. [49]

    1967 Multivariate observations

    MacQueen J. 1967 Multivariate observations. InProceedings ofthe 5th Berkeley symposium on mathematical statisticsand probabilityvol. 1 pp. 281–297. University of California press Oakland, CA, USA

    work page 1967

  50. [50]

    2009 Nonequilibrium Linear Response for Markov Dynamics, I: Jump Processes and Overdamped Diffusions.Journal of Statistical Physics137, 1094–1116

    Baiesi M, Maes C, Wynants B. 2009 Nonequilibrium Linear Response for Markov Dynamics, I: Jump Processes and Overdamped Diffusions.Journal of Statistical Physics137, 1094–1116. (10.1007/s10955-009-9852-8)

    work page doi:10.1007/s10955-009-9852-8 2009

  51. [51]

    Lucarini V . 2016 Response Operators for Markov Processes in a Finite State Space: Radius of Convergence and Link to the Response Theory for Axiom A Systems.Journal of Statistical Physics162, 312–333. (10.1007/s10955-015-1409-4)

    work page doi:10.1007/s10955-015-1409-4 2016

  52. [52]

    2020 Response and Sensitivity Using Markov Chains.Journal of Statistical Physics179, 1572–1593

    Santos Gutiérrez M, Lucarini V . 2020 Response and Sensitivity Using Markov Chains.Journal of Statistical Physics179, 1572–1593. (10.1007/s10955-020-02504-4)

    work page doi:10.1007/s10955-020-02504-4 2020

  53. [53]

    2024 Nonequilibrium Response for Markov Jump Processes: Exact Results and Tight Bounds.Phys

    Aslyamov T, Esposito M. 2024 Nonequilibrium Response for Markov Jump Processes: Exact Results and Tight Bounds.Phys. Rev. Lett.132, 037101. (10.1103/PhysRevLett.132.037101)

    work page doi:10.1103/physrevlett.132.037101 2024

  54. [54]

    2025 Interpretable and Equation-Free Response Theory for Complex Systems.Phil

    Lucarini V . 2025 Interpretable and Equation-Free Response Theory for Complex Systems.Phil. Trans. Roy. Soc. A. (10.1098/rsta.2025.0081)

    work page doi:10.1098/rsta.2025.0081 2025

  55. [55]

    2025 Nonequilibrium Fluctuation-Response Relations: From Identities to Bounds.Phys

    Aslyamov T, Ptaszy ´ nski K, Esposito M. 2025 Nonequilibrium Fluctuation-Response Relations: From Identities to Bounds.Phys. Rev. Lett.134, 157101. (10.1103/PhysRevLett.134.157101) 25royalsocietypublishing.org/journal/rspa Proc R Soc A 0000000

    work page doi:10.1103/physrevlett.134.157101 2025

  56. [56]

    2024MATLAB version 24.1.0.2537033 (R2024a)

    The Mathworks, Inc.. 2024MATLAB version 24.1.0.2537033 (R2024a). Natick, Massachusetts

    work page

  57. [57]

    2023Python 3.12.1 Documentation

    Python Software Foundation. 2023Python 3.12.1 Documentation

    work page

  58. [58]

    Bezanson , author A

    Bezanson J, Edelman A, Karpinski S, Shah VB. 2017 Julia: A Fresh Approach to Numerical Computing.SIAM Review59, 65–98. (10.1137/141000671)

    work page doi:10.1137/141000671 2017

  59. [59]

    2024 Response Theory via Generative Score Modeling.Physical Review Letters133, 267302

    Giorgini LT, Deck K, Bischoff T, Souza AN. 2024 Response Theory via Generative Score Modeling.Physical Review Letters133, 267302. (10.1103/PhysRevLett.133.267302)

    work page doi:10.1103/physrevlett.133.267302 2024

  60. [60]

    Giorgini LT, Falasca F, Souza AN. 2025a Predicting forced responses of probability distributions via the fluctuation–dissipation theorem and generative modeling.Proceedings of the National Academy of Sciences122, e2509578122. (10.1073/pnas.2509578122)

    work page doi:10.1073/pnas.2509578122

  61. [61]

    2025b Statistical Parameter Calibration with the Generalized Fluctuation–Dissipation Theorem and Generative Modeling

    Giorgini LT, Bischoff T, Souza AN. 2025b Statistical Parameter Calibration with the Generalized Fluctuation–Dissipation Theorem and Generative Modeling. arXiv:2509.19660

    work page arXiv

  62. [62]

    1997 Multi-pattern fingerprint method for detection and attribution of climate change.Climate Dynamics13, 601–611

    Hasselmann K. 1997 Multi-pattern fingerprint method for detection and attribution of climate change.Climate Dynamics13, 601–611. (10.1007/s003820050185)

    work page doi:10.1007/s003820050185 1997

  63. [63]

    1999 Checking for model consistency in optimal fingerprinting.Climate Dynamics15, 419–434

    Allen M, Tett S. 1999 Checking for model consistency in optimal fingerprinting.Climate Dynamics15, 419–434. (10.1007/s003820050291)

    work page doi:10.1007/s003820050291 1999

  64. [64]

    2011 Use of models in detection and attribution of climate change.WIREs Climate Change2, 570–591

    Hegerl G, Zwiers F. 2011 Use of models in detection and attribution of climate change.WIREs Climate Change2, 570–591. (https://doi.org/10.1002/wcc.121)

    work page doi:10.1002/wcc.121 2011

  65. [65]

    2014 Optimal fingerprinting under multiple sources of uncertainty.Geophysical Research Letters41, 1261–1268

    Hannart A, Ribes A, Naveau P . 2014 Optimal fingerprinting under multiple sources of uncertainty.Geophysical Research Letters41, 1261–1268. (https://doi.org/10.1002/2013GL058653)

    work page doi:10.1002/2013gl058653 2014

  66. [66]

    2009Probabilistic reasoning in intelligent systems : networks of plausible inference

    Pearl J. 2009Probabilistic reasoning in intelligent systems : networks of plausible inference. San Francisco, Calif.: Morgan Kaufmann. Example for Explaning away

    work page

  67. [67]

    2003 Violation of the fluctuation–dissipation theorem in glassy systems: basic notions and the numerical evidence.Journal of Physics A: Mathematical and General36, R181

    Crisanti A, Ritort F. 2003 Violation of the fluctuation–dissipation theorem in glassy systems: basic notions and the numerical evidence.Journal of Physics A: Mathematical and General36, R181. (10.1088/0305-4470/36/21/201)

    work page doi:10.1088/0305-4470/36/21/201 2003

  68. [68]

    2007 Efficient Measurement of Linear Susceptibilities in Molecular Simulations: Application to Aging Supercooled Liquids.Phys

    Berthier L. 2007 Efficient Measurement of Linear Susceptibilities in Molecular Simulations: Application to Aging Supercooled Liquids.Phys. Rev. Lett.98, 220601. (10.1103/PhysRevLett.98.220601)

    work page doi:10.1103/physrevlett.98.220601 2007

  69. [69]

    2012 Temporal networks.Physics Reports519, 97–125

    Holme P , Saramäki J. 2012 Temporal networks.Physics Reports519, 97–125. Temporal Networks (https://doi.org/10.1016/j.physrep.2012.03.001)

    work page doi:10.1016/j.physrep.2012.03.001 2012

  70. [70]

    2012 Activity driven modeling of time varying networks.Scientific Reports2, 469

    Perra N, Gonçalves B, Pastor-Satorras R, Vespignani A. 2012 Activity driven modeling of time varying networks.Scientific Reports2, 469. (10.1038/srep00469)

    work page doi:10.1038/srep00469 2012

  71. [71]

    & Bianconi, G

    Sun H, Radicchi F, Kurths J, Bianconi G. 2023 The dynamic nature of percolation on networks with triadic interactions.Nature Communications14, 1308. (10.1038/s41467-023-37019-5)

    work page doi:10.1038/s41467-023-37019-5 2023

  72. [72]

    2010 Dynamical and bursty interactions in social networks

    Stehlé J, Barrat A, Bianconi G. 2010 Dynamical and bursty interactions in social networks. Phys. Rev. E81, 035101. (10.1103/PhysRevE.81.035101)

    work page doi:10.1103/physreve.81.035101 2010

  73. [73]

    2024 Triadic percolation induces dynamical topological patterns in higher-order networks.PNAS Nexus3, pgae270

    Millán AP , Sun H, Torres JJ, Bianconi G. 2024 Triadic percolation induces dynamical topological patterns in higher-order networks.PNAS Nexus3, pgae270. (10.1093/pnasnexus/pgae270)

    work page doi:10.1093/pnasnexus/pgae270 2024

  74. [74]

    2021 Consensus dynamics on temporal hypergraphs

    Neuhäuser L, Lambiotte R, Schaub MT. 2021 Consensus dynamics on temporal hypergraphs. Phys. Rev. E104, 064305. (10.1103/PhysRevE.104.064305)

    work page doi:10.1103/physreve.104.064305 2021

  75. [75]

    2021 Time-varying beta, market volatility and stress: A comparison between the United States and India.IIMB Management Review33, 50–63

    Chakrabarti G, Das R. 2021 Time-varying beta, market volatility and stress: A comparison between the United States and India.IIMB Management Review33, 50–63. (https://doi.org/10.1016/j.iimb.2021.03.003)

    work page doi:10.1016/j.iimb.2021.03.003 2021

  76. [76]

    2000 Chaos in periodically forced discrete-time ecosystem models.Chaos, Solitons & Fractals11, 2331–2342

    Summers D, Cranford JG, Healey BP . 2000 Chaos in periodically forced discrete-time ecosystem models.Chaos, Solitons & Fractals11, 2331–2342. (https://doi.org/10.1016/S0960- 0779(99)00154-X)

    work page doi:10.1016/s0960- 2000

  77. [77]

    2000 Periodic Mortality Events in Predator-Prey Systems

    Ives AR, Gross K, Vincent AAJ. 2000 Periodic Mortality Events in Predator-Prey Systems. Ecology81, 3330–3340. (10.2307/177497)

    work page doi:10.2307/177497 2000

  78. [78]

    2024 Partial tipping in bistable ecological systems under periodic environmental variability.Chaos: An Interdisciplinary Journal of Nonlinear Science34, 083130

    Basak A, Dana SK, Bairagi N. 2024 Partial tipping in bistable ecological systems under periodic environmental variability.Chaos: An Interdisciplinary Journal of Nonlinear Science34, 083130. (10.1063/5.0215157)

    work page doi:10.1063/5.0215157 2024

  79. [79]

    2024 Flexible neural population dynamics govern the speed and stability of sensory encoding in mouse visual cortex.Nature Communications15,

    Horrocks EAB, Rodrigues FR, Saleem AB. 2024 Flexible neural population dynamics govern the speed and stability of sensory encoding in mouse visual cortex.Nature Communications15,

    work page 2024

  80. [80]

    (10.1038/s41467-024-50563-y)

    work page doi:10.1038/s41467-024-50563-y

Showing first 80 references.