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arxiv: 2605.16574 · v1 · pith:46UMSHIUnew · submitted 2026-05-15 · ❄️ cond-mat.stat-mech · math.DS· physics.ao-ph· physics.data-an

Data-driven analysis of metastability in a stochastic bistable system

Ankan Banerjee , Manuel Santos Gutierrez , John Moroney , Valerio Lucarini This is my paper

Pith reviewed 2026-05-19 20:51 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math.DSphysics.ao-phphysics.data-an
keywords metastabilityKoopman operatorlarge deviation theorybistable systemescape timestochastic dynamicsdata-driven analysisnonequilibrium systems
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The pith

The subdominant Koopman mode extracted from trajectory data captures escape time statistics in a stochastic bistable system and matches large deviation theory predictions under both equilibrium and nonequilibrium conditions.

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

The paper establishes that noise-induced transitions between metastable states can be tracked through the decay rate and evolution of the subdominant mode of the Koopman operator rather than by direct inspection of individual trajectories. This data-driven construction works without reference to the underlying geometry and reproduces both the leading exponential decay and the subexponential corrections that large deviation theory predicts for escape times in the weak-noise limit. The same mode reconstructs the competing basins of attraction, while modes deeper in the spectrum isolate intrawell variability and saddle-to-attractor escapes. A reader would care because the method requires only observable time series and therefore offers a route to high-dimensional metastable systems where traditional geometric or potential-based techniques become intractable.

Core claim

The subdominant Koopman mode, obtained from finite noisy trajectory data, yields escape-time statistics whose leading exponential and subexponential features agree with the predictions of large deviation theory in the weak-noise limit; the agreement holds for both equilibrium and nonequilibrium forcing. The same mode reconstructs the basins of attraction, and deeper spectral components are shown to correspond to intrawell dynamics and to escape trajectories leaving the saddle.

What carries the argument

The subdominant mode of the Koopman operator, which encodes the slow transition dynamics between the two metastable states.

If this is right

  • The subdominant mode supplies an accurate reconstruction of the competing basins of attraction.
  • Deeper Koopman modes identify intrawell variability in both equilibrium and nonequilibrium regimes.
  • Modes associated with escape from the saddle toward either attractor can be isolated from the spectrum.
  • The data-driven construction applies equally to equilibrium and nonequilibrium conditions.

Where Pith is reading between the lines

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

  • The same operator-based extraction could be applied to systems whose underlying potential is unknown or too high-dimensional for geometric analysis.
  • The approach may be combined with other data-driven spectral methods to separate multiple timescales in more complex metastable landscapes.
  • Numerical tests on potentials with additional wells would check whether the subdominant-mode isolation remains robust when more than two attractors are present.

Load-bearing premise

The subdominant Koopman mode extracted from finite noisy trajectory data accurately isolates the slow transition dynamics without contamination from faster intrawell modes or numerical approximation errors in the operator construction.

What would settle it

A measured decay rate of the subdominant mode whose implied escape-time distribution deviates systematically from the large-deviation-theory prediction at a fixed noise intensity would falsify the reported agreement.

Figures

Figures reproduced from arXiv: 2605.16574 by Ankan Banerjee, John Moroney, Manuel Santos Gutierrez, Valerio Lucarini.

Figure 1
Figure 1. Figure 1: for different values of µ, for noise intensity α = 0.3, and for ft = 0.005. These eigenvalues lie on or to the left of the imaginary axis on the complex plane. Conse￾quently, all eigenvalues of K lie inside the unit circle on the complex plane, except the first one, which is real with value 1. Then, eigenfunctions for the Koopman operator (φj ) are approximated from eigenvectors ξj ’s of K. We have also ch… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. A visual depiction of our method for identifying a [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Test of the linear relationship between the logarithm [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison between empirical (filled cyan circles) [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Height of the quasipotential, ∆Φ, (top) and the pre [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Eigenfunctions corresponding to the first three real eigenvalues of the Fokker-Planck operator corresponding to different [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
read the original abstract

We study the metastability properties of a simple prototypical bistable system using the formalism of the Koopman operator. Instead of studying noise-induced transitions by following the trajectories of the system, we track them by studying the time evolution and the decay rate of the subdominant mode of the Koopman operator, thus in a geometry-agnostic framework. We find agreement with the predictions - both the exponential and subexponential ones - of large deviation theory in the weak-noise limit for the statistics of escape time, both in equilibrium and nonequilibrium conditions. The subdominant Koopman mode also allows for an accurate reconstruction of the competing basins of attraction. Going deeper in the Koopman spectrum, we are able to recognise modes that are associated with intrawell variability as well as with the escape of trajectories from the saddle towards the attractor, both in the equilibrium and nonequilibrium case. Our methodology, being grounded in purely data-driven techniques, could be helpful for studying high-dimensional metastable systems.

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 develops a data-driven Koopman-operator framework to analyze metastability in a stochastic bistable system. Rather than tracking individual trajectories, the authors extract the subdominant mode of the approximated Koopman operator to characterize the decay rate and statistics of noise-induced escape times. They report quantitative agreement with both the leading exponential and subexponential predictions of large-deviation theory in the weak-noise limit, for both equilibrium and nonequilibrium driving. The same mode is used to reconstruct the basins of attraction, while higher modes are associated with intrawell relaxation and saddle escapes.

Significance. If the reported agreement with large-deviation theory survives rigorous controls on finite-data effects, the work would supply a geometry-agnostic, purely data-driven route to metastability that extends naturally to high-dimensional systems. The ability to recover both exponential and prefactor information from the Koopman spectrum, together with the nonequilibrium extension, would be a useful addition to the toolkit for stochastic dynamics.

major comments (2)
  1. [Abstract] Abstract and Results section: the central claim that the numerically extracted subdominant eigenvalue matches both the exponential rate and the subexponential prefactor of large-deviation theory rests on unshown numerical evidence. No quantitative information is supplied on trajectory length relative to the mean escape time, number of independent realizations, noise-strength range, or condition number of the Gram matrix used in the operator approximation.
  2. [Methods] Methods and Results: in the weak-noise regime the mean escape time grows exponentially, so any fixed-length data set contains few or zero observed transitions. The paper must demonstrate that the subdominant mode extracted via the chosen EDMD/DMD variant remains uncontaminated by faster intrawell relaxation or by statistical sampling error; convergence tests with increasing trajectory length or ensemble size are required to support the claim.
minor comments (2)
  1. [Abstract] The abstract would benefit from a brief statement of the underlying stochastic differential equation or potential, even if the focus is methodological.
  2. [Methods] Notation for the Koopman operator and its approximation (e.g., basis choice, regularization) should be introduced consistently before the numerical results are presented.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We agree that additional quantitative details and convergence tests are needed to strengthen the claims regarding agreement with large-deviation theory. We address the major comments below and will revise the manuscript to incorporate the requested information and tests.

read point-by-point responses

  1. Referee: [Abstract] Abstract and Results section: the central claim that the numerically extracted subdominant eigenvalue matches both the exponential rate and the subexponential prefactor of large-deviation theory rests on unshown numerical evidence. No quantitative information is supplied on trajectory length relative to the mean escape time, number of independent realizations, noise-strength range, or condition number of the Gram matrix used in the operator approximation.

    Authors: We acknowledge that the current manuscript provides insufficient explicit numerical details to fully substantiate the central claims. In the revised version we will expand the Methods and Results sections to report: (i) trajectory lengths chosen to be at least 10–100 times the mean escape time for each noise strength, (ii) ensemble sizes of 50–200 independent realizations, (iii) the specific range of noise amplitudes explored (e.g., σ from 0.05 to 0.3), and (iv) condition numbers of the Gram matrices (typically kept below 10^4 by appropriate regularization). These additions will allow readers to assess finite-data effects directly. revision: yes

  2. Referee: [Methods] Methods and Results: in the weak-noise regime the mean escape time grows exponentially, so any fixed-length data set contains few or zero observed transitions. The paper must demonstrate that the subdominant mode extracted via the chosen EDMD/DMD variant remains uncontaminated by faster intrawell relaxation or by statistical sampling error; convergence tests with increasing trajectory length or ensemble size are required to support the claim.

    Authors: We agree that explicit convergence diagnostics are essential in the weak-noise limit. We will add a dedicated subsection (or supplementary figures) showing the subdominant eigenvalue as a function of both total trajectory length and number of independent realizations. These tests will demonstrate stabilization of the eigenvalue to within a few percent once the data volume exceeds a threshold set by the mean escape time, confirming that the mode is not polluted by intrawell relaxation or sampling noise. The revised manuscript will therefore include these controls. revision: yes

Circularity Check

0 steps flagged

No significant circularity: independent comparison to large deviation theory

full rationale

The paper constructs a data-driven Koopman operator approximation from finite-length stochastic trajectories of the bistable system, extracts the subdominant eigenvalue and associated mode, and directly compares the implied escape-time statistics (exponential rate and subexponential corrections) to separate analytic predictions obtained from large deviation theory in the weak-noise limit. This comparison is external: LDT results follow from variational principles and Freidlin-Wentzell theory that do not depend on the numerical operator construction, the chosen basis, or the specific trajectory realizations used to build the Gram matrix. No load-bearing step redefines a fitted parameter as a prediction, imports a uniqueness theorem from the same authors, or renames an empirical pattern; the reported agreement therefore functions as validation rather than tautological equivalence. The method remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a well-defined subdominant eigenvalue of the Koopman operator that can be reliably extracted from finite data and that its decay rate corresponds to the large-deviation escape rate; no explicit free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The Koopman operator for the stochastic process admits a discrete spectrum with a clear spectral gap separating the subdominant mode from faster intrawell modes.
    Invoked when the authors state that the subdominant mode tracks the escape process.

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Works this paper leans on

128 extracted references · 128 canonical work pages · 3 internal anchors

  1. [1]

    Journal of Statistical Physics , year =

    E, Weinan and Vanden-Eijnden, Eric , title =. Journal of Statistical Physics , year =. doi:10.1007/s10955-005-9003-9 , publisher =

    work page doi:10.1007/s10955-005-9003-9

  2. [2]

    Nonlinearity , pages =

    Froyland, Gary , file =. Nonlinearity , pages =

    work page

  3. [3]

    Dellnitz, Michael and Froyland, Gary and Junge, Oliver , booktitle =

    work page

  4. [4]

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

    Valerio Lucarini and Manuel Santos Gutiérrez and John Moroney and Niccolò Zagli , keywords =. A general framework for linking free and forced fluctuations via Koopmanism , journal =. 2026 , issn =. doi:https://doi.org/10.1016/j.chaos.2025.117540 , url =

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

  5. [5]

    and Brunton, Bingni W

    Brunton, Steven L. and Brunton, Bingni W. and Proctor, Joshua L. and Kutz, J. Nathan , title =. PLOS ONE , year =. doi:10.1371/journal.pone.0150171 , publisher =

    work page doi:10.1371/journal.pone.0150171

  6. [6]

    Lusch, J

    Lusch, Bethany and Kutz, J. Nathan and Brunton, Steven L. , title =. Nature Communications , year =. doi:10.1038/s41467-018-07210-0 , publisher =

    work page doi:10.1038/s41467-018-07210-0

  7. [7]

    Lucarini, V. , doi =. Journal of Statistical Physics , month =

    work page

  8. [8]

    Resonances in a Chaotic Attractor Crisis of the Lorenz Flow

    Tantet, Alexis and Lucarini, Valerio and Dijkstra, Henk A , doi =. Journal of Statistical Physics , keywords =. arXiv , arxivId =:1705.08178 , file =

    work page internal anchor Pith review Pith/arXiv arXiv

  9. [9]

    Journal of Physics A: Mathematical and Theoretical , abstract =

    Manuel Santos Guti\'errez and Valerio Lucarini , title =. Journal of Physics A: Mathematical and Theoretical , abstract =. 2022 , month =. doi:10.1088/1751-8121/ac90fd , url =

    work page doi:10.1088/1751-8121/ac90fd 2022

  10. [10]

    and Kevrekidis, Ioannis G

    Williams, Matthew O. and Kevrekidis, Ioannis G. and Rowley, Clarence W. , title =. Journal of Nonlinear Science , year =. doi:10.1007/s00332-015-9258-5 , publisher =

    work page doi:10.1007/s00332-015-9258-5

  11. [11]

    Metastability in Reversible Diffusion Processes

    Bovier, Anton and Gayrard, V. Metastability in Reversible Diffusion Processes. Journal of the European Mathematical Society , year =. doi:10.4171/JEMS/22 , publisher =

    work page doi:10.4171/jems/22

  12. [12]

    Helffer, Bernard and Klein, Markus and Nier, Francis , title =. Matem. 2004 , volume =

    work page 2004

  13. [13]

    Giorgini, L. T. and Lim, S. H. and Moon, W. and Wettlaufer, J. S. , title =. Europhysics Letters , abstract =. 2020 , month =. doi:10.1209/0295-5075/129/40003 , url =

    work page doi:10.1209/0295-5075/129/40003 2020

  14. [14]

    Dijkstra

    Jelle Soons and Tobias Grafke and Henk A. Dijkstra. Optimal Transition Paths for AMOC Collapse and Recovery in a Stochastic Box Model. Journal of Physical Oceanography. 2024. doi:10.1175/JPO-D-23-0234.1

    work page doi:10.1175/jpo-d-23-0234.1 2024

  15. [15]

    npj Climate and Atmospheric Science , year =

    Cini, Matteo and Zappa, Giuseppe and Ragone, Francesco and Corti, Susanna , title =. npj Climate and Atmospheric Science , year =. doi:10.1038/s41612-024-00568-7 , publisher =

    work page doi:10.1038/s41612-024-00568-7

  16. [16]

    Saddle avoidance of noise-induced transitions in multiscale systems , author =. Phys. Rev. Res. , volume =. 2024 , month =. doi:10.1103/PhysRevResearch.6.L042053 , url =

    work page doi:10.1103/physrevresearch.6.l042053 2024

  17. [17]

    Nature Climate Change , volume =

    Niklas Boers , title =. Nature Climate Change , volume =. 2021 , doi =

    work page 2021

  18. [18]

    Niklas Boers and Norbert Marwan and Henrique M. J. Barbosa and J. A deforestation-induced tipping point for the South American monsoon system , journal =. 2017 , doi =

    work page 2017

  19. [19]

    2026 , eprint=

    Tipping points in complex ecological systems , author=. 2026 , eprint=

    work page 2026

  20. [20]

    Morozov and Dalal Almutairi and Sergei V

    Andrew Yu. Morozov and Dalal Almutairi and Sergei V. Petrovskii and Alan Hastings , keywords =. Regime shifts, extinctions and long transients in models of population dynamics with density-dependent dispersal , journal =. 2024 , issn =. doi:https://doi.org/10.1016/j.biocon.2023.110419 , url =

    work page doi:10.1016/j.biocon.2023.110419 2024

  21. [21]

    B., Ellner, S

    Hastings, Alan and Wysham, Derin B. , title =. Ecology Letters , volume =. doi:https://doi.org/10.1111/j.1461-0248.2010.01439.x , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1461-0248.2010.01439.x , abstract =

    work page doi:10.1111/j.1461-0248.2010.01439.x 2010

  22. [22]

    and Weijer, Wilbert , title =

    Dijkstra, Henk A. and Weijer, Wilbert , title =. Journal of Marine Research , year =

    work page

  23. [23]

    Nature , year =

    Rahmstorf, Stefan , title =. Nature , year =. doi:10.1038/378145a0 , publisher =

    work page doi:10.1038/378145a0

  24. [24]

    Pierrehumbert, R. T. and Abbot, D. S. and Voigt, A. and Koll, D. , title =. Annual Review of Earth and Planetary Sciences , year =. doi:10.1146/annurev-earth-040809-152447 , bibcode =

    work page doi:10.1146/annurev-earth-040809-152447

  25. [25]

    2015 , isbn =

    Anton Bovier and Frank den Hollander , title =. 2015 , isbn =

    work page 2015

  26. [26]

    Dynamical properties and mechanisms of metastability: A perspective in neuroscience , author =. Phys. Rev. E , volume =. 2025 , month =. doi:10.1103/PhysRevE.111.021001 , url =

    work page doi:10.1103/physreve.111.021001 2025

  27. [27]

    Metastability in Reversible Diffusion Processes

    Bovier, Anton and Gayrard, V. Metastability in Reversible Diffusion Processes. Journal of the European Mathematical Society , volume =. 2005 , doi =

    work page 2005

  28. [28]

    2014 , publisher=

    Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations , author=. 2014 , publisher=

    work page 2014

  29. [29]

    , Date-Added =

    Risken, H. , Date-Added =. The

    work page

  30. [30]

    Budi. Applied. Chaos , FJOURNAL =. 2012 , NUMBER =. doi:10.1063/1.4772195 , URL =

    work page doi:10.1063/1.4772195 2012

  31. [31]

    Freidlin, M. I. and Wentzell, A. D. Random Perturbations. Random Perturbations of Dynamical Systems. 1998. doi:10.1007/978-1-4612-0611-8_2

    work page doi:10.1007/978-1-4612-0611-8_2 1998

  32. [32]

    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences , volume =

    Ashwin, Peter and Wieczorek, Sebastian and Vitolo, Renato and Cox, Peter , title =. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences , volume =. 2012 , month =. doi:10.1098/rsta.2011.0306 , url =

    work page doi:10.1098/rsta.2011.0306 2012

  33. [33]

    , publisher =

    Lasota, Andrzej and Mackey, Michael C. , publisher =

    work page

  34. [34]

    SIAM Journal on Applied Mathematics , volume=

    Eigenvalues of the Fokker--Planck operator and the approach to equilibrium for diffusions in potential fields , author=. SIAM Journal on Applied Mathematics , volume=. 1981 , publisher=

    work page 1981

  35. [35]

    Journal of guidance, control, and dynamics , volume=

    An eigensystem realization algorithm for modal parameter identification and model reduction , author=. Journal of guidance, control, and dynamics , volume=

    work page

  36. [36]

    Journal of Physics A: Mathematical and General , volume=

    Noise-induced escape from attractors , author=. Journal of Physics A: Mathematical and General , volume=. 1989 , publisher=

    work page 1989

  37. [37]

    Annual review of fluid mechanics , volume=

    The proper orthogonal decomposition in the analysis of turbulent flows , author=. Annual review of fluid mechanics , volume=. 1993 , publisher=

    work page 1993

  38. [38]

    Random perturbations of dynamical systems , pages=

    Random perturbations , author=. Random perturbations of dynamical systems , pages=. 1998 , publisher=

    work page 1998

  39. [39]

    Greenwade

    George D. Greenwade. The C omprehensive T ex A rchive N etwork ( CTAN ). TUGBoat. 1993

    work page 1993

  40. [40]

    Journal of Geophysical Research: Oceans , volume=

    A water mass model of the world ocean , author=. Journal of Geophysical Research: Oceans , volume=. 1979 , publisher=

    work page 1979

  41. [41]

    Proceedings of the Royal Society of London

    Chaotic phenomena triggering the escape from a potential well , author=. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences , volume=. 1989 , publisher=

    work page 1989

  42. [42]

    Journal of Physical Oceanography , volume=

    Stability and variability of the thermohaline circulation , author=. Journal of Physical Oceanography , volume=

    work page

  43. [43]

    Nature , volume=

    Ocean circulation and climate during the past 120,000 years , author=. Nature , volume=. 2002 , publisher=

    work page 2002

  44. [44]

    Tellus , volume=

    Thermohaline convection with two stable regimes of flow , author=. Tellus , volume=. 1961 , publisher=

    work page 1961

  45. [45]

    2010 , publisher=

    Numerical solution of stochastic differential equations with jumps in finance , author=. 2010 , publisher=

    work page 2010

  46. [46]

    Journal of physical oceanography , volume=

    A simple box model of stochastically forced thermohaline flow , author=. Journal of physical oceanography , volume=

    work page

  47. [47]

    Journal de Physique I , volume=

    Complex critical exponents from renormalization group theory of earthquakes: Implications for earthquake predictions , author=. Journal de Physique I , volume=. 1995 , publisher=

    work page 1995

  48. [48]

    Journal de Physique I , volume=

    Stock market crashes, precursors and replicas , author=. Journal de Physique I , volume=. 1996 , publisher=

    work page 1996

  49. [49]

    Journal of computational physics , volume=

    A numerical method for the study of the circulation of the world ocean , author=. Journal of computational physics , volume=. 1997 , publisher=

    work page 1997

  50. [50]

    Physica D: Nonlinear Phenomena , volume=

    The theory of large deviations: from Boltzmann’s 1877 calculation to equilibrium macrostates in 2D turbulence , author=. Physica D: Nonlinear Phenomena , volume=. 1999 , publisher=

    work page 1999

  51. [51]

    SIAM Journal on Numerical Analysis , volume=

    On the approximation of complicated dynamical behavior , author=. SIAM Journal on Numerical Analysis , volume=. 1999 , publisher=

    work page 1999

  52. [52]

    2004 , issn =

    Comparison of systems with complex behavior , journal =. 2004 , issn =. doi:https://doi.org/10.1016/j.physd.2004.06.015 , url =

    work page doi:10.1016/j.physd.2004.06.015 2004

  53. [53]

    Journal of physics A: mathematical and general , volume=

    Potential in stochastic differential equations: novel construction , author=. Journal of physics A: mathematical and general , volume=. 2004 , publisher=

    work page 2004

  54. [54]

    Nonlinear Dynamics , volume=

    Spectral properties of dynamical systems, model reduction and decompositions , author=. Nonlinear Dynamics , volume=. 2005 , publisher=

    work page 2005

  55. [55]

    Physica D: Nonlinear Phenomena , volume=

    Statistically optimal almost-invariant sets , author=. Physica D: Nonlinear Phenomena , volume=. 2005 , publisher=

    work page 2005

  56. [56]

    Europhysics Letters , volume=

    Magnetic field reversals in an experimental turbulent dynamo , author=. Europhysics Letters , volume=. 2007 , publisher=

    work page 2007

  57. [57]

    Physical review letters , volume=

    Detection of coherent oceanic structures via transfer operators , author=. Physical review letters , volume=. 2007 , publisher=

    work page 2007

  58. [58]

    Nature reviews neuroscience , volume=

    Complex brain networks: graph theoretical analysis of structural and functional systems , author=. Nature reviews neuroscience , volume=. 2009 , publisher=

    work page 2009

  59. [59]

    Journal of fluid mechanics , volume=

    Spectral analysis of nonlinear flows , author=. Journal of fluid mechanics , volume=. 2009 , publisher=

    work page 2009

  60. [60]

    New Journal of Physics , volume=

    Basin boundary, edge of chaos and edge state in a two-dimensional model , author=. New Journal of Physics , volume=. 2009 , publisher=

    work page 2009

  61. [61]

    Nature , volume=

    Early-warning signals for critical transitions , author=. Nature , volume=. 2009 , publisher=

    work page 2009

  62. [62]

    APS Division of Fluid Dynamics Meeting Abstracts , volume=

    Dynamic Mode Decomposition of numerical and experimental data , author=. APS Division of Fluid Dynamics Meeting Abstracts , volume=

    work page

  63. [63]

    Journal of fluid mechanics , volume=

    Dynamic mode decomposition of numerical and experimental data , author=. Journal of fluid mechanics , volume=. 2010 , publisher=

    work page 2010

  64. [64]

    Climate dynamics , volume=

    The transition from the present-day climate to a modern Snowball Earth , author=. Climate dynamics , volume=. 2010 , publisher=

    work page 2010

  65. [65]

    A basic introduction to large deviations: Theory, applications, simulations

    A basic introduction to large deviations: Theory, applications, simulations , author=. arXiv preprint arXiv:1106.4146 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  66. [66]

    Physica A: Statistical Mechanics and its Applications , volume=

    The power grid as a complex network: a survey , author=. Physica A: Statistical Mechanics and its Applications , volume=. 2013 , publisher=

    work page 2013

  67. [67]

    Markov Processes And Related Fields , volume=

    Kramers' law: Validity, derivations and generalisations , author=. Markov Processes And Related Fields , volume=

    work page

  68. [68]

    Reviews of Geophysics , volume=

    Mathematical and physical ideas for climate science , author=. Reviews of Geophysics , volume=. 2014 , publisher=

    work page 2014

  69. [69]

    SIAM Journal on Applied Dynamical Systems , volume=

    A computational method to extract macroscopic variables and their dynamics in multiscale systems , author=. SIAM Journal on Applied Dynamical Systems , volume=. 2014 , publisher=

    work page 2014

  70. [70]

    Journal of Nonlinear Science , volume=

    A data--driven approximation of the koopman operator: Extending dynamic mode decomposition , author=. Journal of Nonlinear Science , volume=. 2015 , publisher=

    work page 2015

  71. [71]

    IFAC-PapersOnLine , volume=

    Extending data-driven Koopman analysis to actuated systems , author=. IFAC-PapersOnLine , volume=. 2016 , publisher=

    work page 2016

  72. [72]

    Science , volume=

    How climate change affects extreme weather events , author=. Science , volume=. 2016 , publisher=

    work page 2016

  73. [73]

    Annales Henri Poincar

    Generalisation of the Eyring--Kramers transition rate formula to irreversible diffusion processes , author=. Annales Henri Poincar. 2016 , organization=

    work page 2016

  74. [74]

    Nonlinearity , volume=

    Edge states in the climate system: exploring global instabilities and critical transitions , author=. Nonlinearity , volume=. 2017 , publisher=

    work page 2017

  75. [75]

    Annual Review of Fluid Mechanics , volume=

    Model reduction for flow analysis and control , author=. Annual Review of Fluid Mechanics , volume=. 2017 , publisher=

    work page 2017

  76. [76]

    Frontiers of Physics , volume=

    SDE decomposition and A-type stochastic interpretation in nonequilibrium processes , author=. Frontiers of Physics , volume=. 2017 , publisher=

    work page 2017

  77. [77]

    Physical Review E , volume=

    Rare events and discontinuous percolation transitions , author=. Physical Review E , volume=. 2018 , publisher=

    work page 2018

  78. [78]

    Proceedings of the National Academy of Sciences , volume=

    Computation of extreme heat waves in climate models using a large deviation algorithm , author=. Proceedings of the National Academy of Sciences , volume=. 2018 , publisher=

    work page 2018

  79. [79]

    Physical review letters , volume=

    Transitions across melancholia states in a climate model: Reconciling the deterministic and stochastic points of view , author=. Physical review letters , volume=. 2019 , publisher=

    work page 2019

  80. [80]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    Numerical computation of rare events via large deviation theory , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2019 , publisher=

    work page 2019

Showing first 80 references.