pith. sign in

arxiv: 2106.01160 · v3 · submitted 2021-06-02 · 🧮 math.DS · math.CA· math.PR· nlin.PS

A General View on Double Limits in Differential Equations

Christian Kuehn , Nils Berglund , Christian Bick , Maximilian Engel , Tobias Hurth , Annalisa Iuorio , Cinzia Soresina This is my paper

Pith reviewed 2026-05-24 12:37 UTC · model grok-4.3

classification 🧮 math.DS math.CAmath.PRnlin.PS
keywords double limitssingular perturbationsdifferential equationsmultiple time scalesstochastic dynamicsparameter spaceasymptotic analysisnetwork coupling
0
0 comments X

The pith

A three-step process unifies the study of double-limit problems in differential equations by partitioning parameter space near singular limits.

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

The paper proposes a conceptual framework to compare singularly perturbed differential equations that involve multiple small parameters. The framework begins by specifying the problem setting and identifying the small parameters, then defines a notion of equivalence based on a chosen property or observable to divide the parameter space into regions, and finally examines the asymptotic singular limit problems that arise in those regions. This sequence is first shown on basic algebraic and analytic examples and then applied to contemporary cases such as multiple time scales, stochastic dynamics, spatial patterns, and network coupling. A sympathetic reader would care because the approach supplies a consistent diagrammatic template that makes results from otherwise separate literatures easier to place side by side and compare.

Core claim

The authors develop a general conceptual framework for double-limit problems in singularly perturbed differential equations. The framework consists of specifying the problem setting and small parameters, defining a notion of equivalence via a property or observable to partition the parameter space, and studying the possible asymptotic singular limit problems together with perturbation results. When this three-step process is carried out on examples from multiple time scales, stochastic dynamics, spatial patterns, and network coupling, the resulting parametric diagrams already provide an excellent unifying theme across the different classes of problems.

What carries the argument

The three-step process of specifying settings and small parameters, defining equivalence via an observable to partition parameter space, and analyzing asymptotic singular limits to complete the subdivision.

If this is right

  • The process produces comparable diagrammatic subdivisions for multiple-time-scale problems.
  • Stochastic dynamics examples fit the same partitioning scheme.
  • Spatial pattern formation and network coupling problems admit the same three-step treatment.
  • Common structural features appear across all reviewed classes once the diagrams are drawn.
  • The methodology supplies a template that can be transferred to other classes of differential equations.

Where Pith is reading between the lines

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

  • The same template might be used to organize double-limit questions that arise in numerical analysis or data-driven modeling.
  • Software that automates the equivalence-partition step could be built once the observable is chosen.
  • The framework could be tested on a problem that combines stochastic effects with spatial patterns to check whether the diagrams remain legible.
  • Explicit comparison of the observables chosen in each reviewed example might reveal deeper algebraic relations among the limit problems.

Load-bearing premise

The three-step process can be applied uniformly and insightfully to diverse modern examples without requiring substantial problem-specific modifications.

What would settle it

A new double-limit problem in differential equations to which the three-step process cannot be applied without major custom changes or that yields no clear partition of the parameter space would falsify the unifying claim.

Figures

Figures reproduced from arXiv: 2106.01160 by Annalisa Iuorio, Christian Bick, Christian Kuehn, Cinzia Soresina, Maximilian Engel, Nils Berglund, Tobias Hurth.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Classification diagram with respect to the property [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Classification diagram with respect to the property [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Classification diagram with respect to the property [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Classification diagram with respect to the property [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Classification diagram with respect to the property [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Probability [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Slow passage through a transcritical bifurcation. The [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: The probability [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: ( [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Phase space of the stochastic FitzHugh–Nagumo [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Time series [PITH_FULL_IMAGE:figures/full_fig_p011_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Pullback attraction to random equilibrium (a)-(c) [PITH_FULL_IMAGE:figures/full_fig_p012_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: Fixing [PITH_FULL_IMAGE:figures/full_fig_p012_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: Fixing all other parameters in model ( [PITH_FULL_IMAGE:figures/full_fig_p013_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: Sample trajectories for the vector fields [PITH_FULL_IMAGE:figures/full_fig_p014_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19: Classification diagram with respect to the property [PITH_FULL_IMAGE:figures/full_fig_p015_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20: Classification diagram with respect to the property [PITH_FULL_IMAGE:figures/full_fig_p016_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21: (a) Numerically computed bifurcation diagram of [PITH_FULL_IMAGE:figures/full_fig_p017_21.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23: Covering of the bifurcation diagram for ( [PITH_FULL_IMAGE:figures/full_fig_p019_23.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22: Singular solutions to ( [PITH_FULL_IMAGE:figures/full_fig_p019_22.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24: Classification diagram of ( [PITH_FULL_IMAGE:figures/full_fig_p019_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25: Schematic representation of the systems of PDEs [PITH_FULL_IMAGE:figures/full_fig_p021_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26: Bifurcation diagrams with respect to the bifurcation [PITH_FULL_IMAGE:figures/full_fig_p022_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: FIG. 27: Qualitative classification diagram of system ( [PITH_FULL_IMAGE:figures/full_fig_p023_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: FIG. 28 [PITH_FULL_IMAGE:figures/full_fig_p024_28.png] view at source ↗
read the original abstract

In this paper, we review several results from singularly perturbed differential equations with multiple small parameters. In addition, we develop a general conceptual framework to compare and contrast the different results by proposing a three-step process. First, one specifies the setting and restrictions of the differential equation problem to be studied and identifies the relevant small parameters. Second, one defines a notion of equivalence via a property/observable for partitioning the parameter space into suitable regions near the singular limit. Third, one studies the possible asymptotic singular limit problems as well as perturbation results to complete the diagrammatic subdivision process. We illustrate this approach for two simple problems from algebra and analysis. Then we proceed to the review of several modern double-limit problems including multiple time scales, stochastic dynamics, spatial patterns, and network coupling. For each example, we illustrate the previously mentioned three-step process and show that already double-limit parametric diagrams provide an excellent unifying theme. After this review, we compare and contrast the common features among the different examples. We conclude with a brief outlook, how our methodology can help to systematize the field better, and how it can be transferred to a wide variety of other classes of differential equations.

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

0 major / 2 minor

Summary. The paper claims that a three-step process—specifying the setting and small parameters, defining equivalence via a property/observable for partitioning parameter space, and studying asymptotic singular limit problems—provides an excellent unifying theme for double-limit problems in differential equations. It illustrates this with two simple problems from algebra and analysis, then applies the process to review examples in multiple time scales, stochastic dynamics, spatial patterns, and network coupling. For each, it performs the partitioning and diagrammatic subdivision, followed by a comparison of common features and an outlook on systematizing the field.

Significance. The proposed framework supplies a consistent organizational lens that makes the partitioning of parameter spaces and the resulting limit problems explicit across disparate classes of singularly perturbed systems. By carrying out the three steps uniformly on both elementary examples and modern applications, the manuscript demonstrates a practical unifying theme that can aid comparison of results; the explicit diagrammatic treatment of each class is a concrete strength that supports the claim of improved systematization.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'diagrammatic subdivision process' appears without prior definition; a one-sentence gloss would improve accessibility for readers encountering the framework for the first time.
  2. [Comparison section] Comparison section: while common features are discussed, a compact table listing the chosen observable/equivalence relation and the resulting partition for each of the four modern examples would make the unifying theme easier to grasp at a glance.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; framework is organizational and externally illustrated

full rationale

The paper proposes a three-step conceptual process (specify setting and small parameters; define equivalence/observable for partitioning; study asymptotic limits and perturbations) as an organizing theme for double-limit problems. It illustrates the process first on independent algebra and analysis examples, then applies the same subdivision to external modern cases (multiple time scales, stochastic dynamics, spatial patterns, network coupling) drawn from the literature. No derivation reduces a claimed result to its own fitted inputs, self-definitional equations, or a load-bearing self-citation chain; the central claim is simply that the diagrammatic subdivision can be performed uniformly, which the manuscript demonstrates by construction without internal reduction. This is a standard review-style contribution whose validity rests on the external examples rather than self-referential definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a review and framework proposal that relies on standard background results from differential equation theory without introducing new free parameters or invented entities.

axioms (1)
  • standard math Standard assumptions of differential equation theory such as local existence and uniqueness of solutions under Lipschitz conditions.
    The paper works with differential equations and singular perturbations, so it inherits these background results.

pith-pipeline@v0.9.0 · 5762 in / 1174 out tokens · 37091 ms · 2026-05-24T12:37:46.230110+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

228 extracted references · 228 canonical work pages · 1 internal anchor

  1. [1]

    (30) In terms of matched asymptotics, such sections describe the transition between outer and inner regions

    Translating Σ out 1 in terms of K2- coordinates, we obtain the section Σin 2 := { (σ−1,y 2,ξ 2,r 2) ⏐⏐y2∈ [y−,y +], ξ2∈ [ξ−,ξ +], r2∈ [0,ρσ ]}. (30) In terms of matched asymptotics, such sections describe the transition between outer and inner regions. In par- ticular, the outer regime corresponds to the area limited by Σin 1 and Σout 1 in K1, while the i...

    work page

  2. [2]

    δ∈ [√ε, 2√ 3 ] ; see Figure 24

    Hence, in these regions, we need to take λ∈ [3 4ε, 1 ] , i.e. δ∈ [√ε, 2√ 3 ] ; see Figure 24. The two main parameters we consider in our proofs are hereε and λ ∥u∥2 2 R1 R2 R3 B1 B2 B3 R3 B3R2 R1 B1 B2 λ ∥u∥2 2 FIG. 23: Covering of the bifurcation diagram for ( Xmes) by regionsR1 (brown),R2 (pink), and R3 (magenta). The branches of solutions to (Xmes) for...

    work page

  3. [3]

    compression

    The proof consists of two parts: we first consider a small neighborhood of δ∗ = 2√ 3, i.e. of λ = 3 4ε, where the saddle-node bifur- cation occurs. We define a suitable bifurcation equation, which describes the transition from solutions which limit on type M1-solutions to those which limit on solutions of type M2. Such equation is constructed by imposing th...

    work page

  4. [4]

    typical” or “common

    COMP ARISON In Section 3, we have described a wide variety of doubly-singular limit problems arising in differential equations. Yet, from the different examples, several themes emerge for the future of multiple singular limit systems. Property Types: We have seen various ways of defin- ing propertiesP to obtain double limits which, however, share quite surpr...

    work page

  5. [5]

    singular

    OUTLOOK In this review, we have only been able to illustrate a more general framework for differential equations with multiple small parameters for certain classes of prob- lems. It is evident that many important questions still re- main. To illustrate the diversity of remaining problems, we present a few crucial questions that seem tractable within the ne...

    work page 2046

  6. [6]

    Arnaudon, A

    A. Arnaudon, A. L. De Castro, and D. D. Holm. Noise and dissipation on coadjoint orbits. J. Nonlinear Sci. , 28(1):91–145, 2018

    work page 2018

  7. [7]

    L. Arnold. Random Dynamical Systems . Springer, Berlin, 1998

    work page 1998

  8. [8]

    Arnold, N

    L. Arnold, N. Sri Namachchivaya, and K.R. Schenk- Hopp´ e. Toward an understanding of stochastic Hopf bifurcation: a case study. Internat. J. Bifur. Chaos Appl. Sci. Engrg. , 6(11):1947–1975, 1996

    work page 1947

  9. [9]

    Arrhenius

    S. Arrhenius. On the reaction velocity of the inversion of cane sugar by acids. J. Phys. Chem. , 4:226, 1889. In German. Translated and published in: Selected Read- ings in Chemical Kinetics, M.H. Back and K.J. Laider (eds.), Pergamon, Oxford, 1967

    work page 1967

  10. [10]

    Ashwin and J.W

    P. Ashwin and J.W. Swift. The dynamics of n weakly coupled identical oscillators. J. Nonlinear Sci., 2(1):69– 108, 1992

    work page 1992

  11. [11]

    State-dependent effective interactions in oscillator net- works through coupling functions with dead zones

    Peter Ashwin, Christian Bick, and Camille Poignard. State-dependent effective interactions in oscillator net- works through coupling functions with dead zones. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences , 377(2160):20190042, 2019

    work page 2019

  12. [12]

    Mathematical Frameworks for Oscillatory Network Dy- namics in Neuroscience

    Peter Ashwin, Stephen Coombes, and Rachel Nicks. Mathematical Frameworks for Oscillatory Network Dy- namics in Neuroscience. The Journal of Mathematical Neuroscience, 6(1):2, 2016

    work page 2016

  13. [13]

    Hopf normal form with S N symmetry and reduction to systems of nonlin- early coupled phase oscillators

    Peter Ashwin and Ana Rodrigues. Hopf normal form with S N symmetry and reduction to systems of nonlin- early coupled phase oscillators. Physica D, 325:14–24, 2016

    work page 2016

  14. [14]

    Bakhtin and Tobias H

    Y. Bakhtin and Tobias H. Invariant densities for dy- namical systems with random switching. Nonlinearity, 25(10):2937–2952, 2012

    work page 2012

  15. [15]

    Bakhtin, T

    Y. Bakhtin, T. Hurth, S.D. Lawley, and J.C. Mattingly. Singularities of invariant densities for random switching between two linear odes in 2D. arXiv:2009.01299, 2020

    work page arXiv 2009

  16. [16]

    Bakhtin, T

    Y. Bakhtin, T. Hurth, and J.C. Mattingly. Regularity of invariant densities for 1d-systems with random switch- ing. Nonlinearity, 28:3755–3787, 2015

    work page 2015

  17. [17]

    Balde, U

    M. Balde, U. Boscain, and P. Mason. A note on stabil- ity conditions for planar switched systems. Internat. J. Control, 82(10):1882–1888, 2009

    work page 2009

  18. [18]

    Networks beyond pairwise interactions: Structure and dynamics

    Federico Battiston, Giulia Cencetti, Iacopo Iacopini, Vito Latora, Maxime Lucas, Alice Patania, Jean-gabriel Young, and Giovanni Petri. Networks beyond pairwise interactions: Structure and dynamics. Physics Reports, 874:1–92, 2020

    work page 2020

  19. [19]

    Baxendale and Priscilla E

    Peter H. Baxendale and Priscilla E. Greenwood. Sus- tained oscillations for density dependent Markov pro- cesses. J. Math. Biol. , 63(3):433–457, 2011

    work page 2011

  20. [20]

    Baxendale

    P.H. Baxendale. A stochastic Hopf bifurcation. Probab. Theory Related Fields, 99(4):581–616, 1994

    work page 1994

  21. [21]

    Baxendale

    P.H. Baxendale. Lyapunov exponents and stability for the stochastic Duffing-van der Pol oscillator. In IUTAM Symposium on Nonlinear Stochastic Dynamics , volume 110 of Solid Mech. Appl. , pages 125–135. Kluwer Acad. Publ., Dordrecht, 2003

    work page 2003

  22. [22]

    Baxendale

    P.H. Baxendale. Stochastic averaging and asymptotic behavior of the stochastic Duffing-van der Pol equation. Stochastic Process. Appl., 113(2):235–272, 2004

    work page 2004

  23. [23]

    Bena¨ ım

    M. Bena¨ ım. Stochastic persistence (part I). Available at https://arxiv.org/abs/1806.08450, 2018. preprint

    work page arXiv 2018

  24. [24]

    Bena¨ ım, S

    M. Bena¨ ım, S. Le Borgne, F. Malrieu, and P.-A. Zitt. On the stability of planar randomly switched systems. Ann. Appl. Probab., 24(1):292–311, 2014

    work page 2014

  25. [25]

    Bena¨ ım, S

    M. Bena¨ ım, S. Le Borgne, F. Malrieu, and P.-A. Zitt. Qualitative properties of certain piecewise deterministic Markov processes. Ann. Inst. Henri Poincar´ e Probab. Stat., 51(3):1040–1075, 2015

    work page 2015

  26. [26]

    A user-friendly condition for exponential ergodicity in randomly switched environments

    Michel Bena¨ ım, Tobias Hurth, and Edouard Strickler. A user-friendly condition for exponential ergodicity in randomly switched environments. Electron. Commun. Probab., 23:1–12, 2018

    work page 2018

  27. [27]

    Quantitative ergodicity for some switched dynamical systems

    Michel Bena¨ ım, St´ ephane Le Borgne, Florent Malrieu, and Pierre-Andr´ e Zitt. Quantitative ergodicity for some switched dynamical systems. Electron. Commun. Probab., 17:no. 56, 14, 2012

    work page 2012

  28. [28]

    Random switch- ing between vector fields having a common zero

    Michel Bena¨ ım and Edouard Strickler. Random switch- ing between vector fields having a common zero. Ann. Appl. Probab., 29(1):326–375, 2019

    work page 2019

  29. [29]

    how switch- ing between beneficial environments can make survival harder

    Michel Bena¨ ım and Claude Lobry. Lotka–volterra with randomly fluctuating environments or “how switch- ing between beneficial environments can make survival harder”. Ann. Appl. Probab., 26(6):3754–3785, 12 2016

    work page 2016

  30. [30]

    Bender and S.A

    C.M. Bender and S.A. Orszag. Asymptotic Methods and Perturbation Theory. Springer, 1999

    work page 1999

  31. [31]

    Benoˆ ıt, J.L

    E. Benoˆ ıt, J.L. Callot, F. Diener, and M. Diener. Chasse au canards. Collect. Math., 31:37–119, 1981

    work page 1981

  32. [32]

    Bensoussan, J.-L

    A. Bensoussan, J.-L. Lions, and G. Papanicolaou. Asymptotic analysis for periodic structures . Chelsea, 2011

    work page 2011

  33. [33]

    Berardo, S

    C. Berardo, S. Geritz, M. Gyllenberg, and G. Raoul. Interactions between different predator–prey states: a method for the derivation of the functional and numer- ical response. J. Math. Biol. , 80:2431–2468, 2020

    work page 2020

  34. [34]

    Berglund and B

    N. Berglund and B. Gentz. Pathwise description of dynamic pitchfork bifurcations with additive noise. Probab. Theory Rel., 122(3):341–388, 2002

    work page 2002

  35. [35]

    Berglund and B

    N. Berglund and B. Gentz. A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential. Ann. Appl. Probab. , 12:1419– 1470, 2002

    work page 2002

  36. [36]

    Berglund and B

    N. Berglund and B. Gentz. Geometric singular per- turbation theory for stochastic differential equations. J. Differ. Equations , 191:1–54, 2003

    work page 2003

  37. [37]

    Berglund and B

    N. Berglund and B. Gentz. Noise-induced phenomena in slow–fast dynamical systems. A sample-paths approach . Probability and its Applications. Springer-Verlag, Lon- don, 2006

    work page 2006

  38. [38]

    Berglund, B

    N. Berglund, B. Gentz, and C. Kuehn. Hunting French ducks in a noisy environment. J. Differ. Equations , 252(9):4786–4841, 2012

    work page 2012

  39. [39]

    Berglund, B

    N. Berglund, B. Gentz, and C. Kuehn. From random Poincar´ e maps to stochastic mixed-mode-oscillation patterns. J. Dyn. Differ. Equ. , 27(1):83–136, 2015

    work page 2015

  40. [40]

    Berglund and D

    N. Berglund and D. Landon. Mixed-mode oscilla- tions and interspike interval statistics in the stochas- tic FitzHugh-Nagumo model. Nonlinearity, 25(8):2303– 2335, 2012

    work page 2012

  41. [41]

    Berner, E

    R. Berner, E. Sch¨ oll, and S. Yanchuk. Multiclusters in networks of adaptively coupled phase oscillators. SIAM 29 J. Appl. Dyn. Sys. , 18(4):2227–2266, 2019

    work page 2019

  42. [42]

    G. Beutler. Methods of Celestial Mechanics (volume I): physical, mathematical, and numerical principles . Springer, 2004

    work page 2004

  43. [43]

    C. Bick, P. Ashwin, and A. Rodrigues. Chaos in generi- cally coupled phase oscillator networks with nonpairwise interactions. Chaos, 26(9):094814, 2016

    work page 2016

  44. [44]

    C. Bick, M. Timme, D. Paulikat, D. Rathlev, and P. Ashwin. Chaos in symmetric phase oscillator net- works. Phys. Rev. Lett., 107(24):244101, 2011

    work page 2011

  45. [45]

    Laing, and Erik A

    Christian Bick, Marc Goodfellow, Carlo R. Laing, and Erik A. Martens. Understanding the dynamics of bi- ological and neural oscillator networks through exact mean-field reductions: a review. The Journal of Math- ematical Neuroscience, 10(1):9, 2020

    work page 2020

  46. [46]

    Harring- ton, and Michael T

    Christian Bick, Elizabeth Gross, Heather A. Harring- ton, and Michael T. Schaub. What are higher-order networks? arXiv:2104.11329, apr 2021

    work page arXiv 2021

  47. [47]

    Blackbeard, H

    N. Blackbeard, H. Erzgr¨ aber, and S. Wieczorek. Shear- induced bifurcations and chaos in models of three cou- pled lasers. SIAM J. Appl. Dyn. Syst. , 10(2):469–509, 2011

    work page 2011

  48. [48]

    Blumenthal, J

    A. Blumenthal, J. Xue, and L.-S. Young. Lyapunov exponents for random perturbations of some area- preserving maps including the standard map. Ann. of Math. (2), 185(1):285–310, 2017

    work page 2017

  49. [49]

    Blumenthal, J

    A. Blumenthal, J. Xue, and L.-S. Young. Lyapunov ex- ponents and correlation decay for random perturbations of some prototypical 2D maps. Comm. Math. Phys. , 359(1):347–373, 2018

    work page 2018

  50. [50]

    Bodnar and J.J.L

    M. Bodnar and J.J.L. Vel´ azquez. An integro-differential equation arising as a limit of individual cell-based mod- els. J. Differen. Equat. , 222(2):341–380, 2006

    work page 2006

  51. [51]

    Borkar and S.K

    V.S. Borkar and S.K. Mitter. A strong approximation theorem for stochastic recursive algorithms. J. Optim. Theor. Appl., 100(3):499–513, 1999

    work page 1999

  52. [52]

    Characterizing mixed mode oscillations shaped by noise and bifurcation structure

    Peter Borowski, Rachel Kuske, Yue-Xian Li, and Juan Luis Cabrera. Characterizing mixed mode oscillations shaped by noise and bifurcation structure. Chaos, 20(4):043117, 2010

    work page 2010

  53. [53]

    Bossolini, M

    E. Bossolini, M. Brøns, and K.U. Kristiansen. A stiction oscillator with canards: on piecewise smooth nonuniqueness and its resolution by regularization us- ing geometric singular perturbation theory. SIAM Rev., 62(4):869–897, 2020

    work page 2020

  54. [54]

    Bothe and D

    D. Bothe and D. Hilhorst. A reaction–diffusion sys- tem with fast reversible reaction. J. Math. Anal. Appl. , 286(1):125–135, 2003

    work page 2003

  55. [55]

    Bothe and M

    D. Bothe and M. Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reac- tion. Discrete Contin. Dyn. Syst. Ser. S , 5(1):49, 2012

    work page 2012

  56. [56]

    Boudjellaba and T

    H. Boudjellaba and T. Sari. Dynamic transcritical bi- furcations in a class of slow-fast predator-prey models. J. Diff. Eq. , 246:2205–2225, 2009

    work page 2009

  57. [57]

    Breden and M

    M. Breden and M. Engel. Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems. arXiv:2101.01491, pages 1–39, 2020

    work page arXiv 2020

  58. [58]

    Breden, C

    M. Breden, C. Kuehn, and C. Soresina. On the influ- ence of cross-diffusion in pattern formation. J. Comput. Dyn., 8(2):213–240, 2021

    work page 2021

  59. [59]

    Brocchieri, L

    E. Brocchieri, L. Corrias, H. Dietert, and Y.-J. Kim. Evolution of dietary diversity and a starvation driven cross-diffusion system as its singular limit. arXiv preprint arXiv:2011.10304, 2020

    work page arXiv 2011

  60. [60]

    Buzzi, P.R

    C.A. Buzzi, P.R. da Silva, and M.A. Teixeira. A singular approach to discontinuous vector fields on the plane. J. Diff. Eq., 231:633–655, 2006

    work page 2006

  61. [61]

    Cardin and M.A

    P.T. Cardin and M.A. Teixeira. Fenichel theory for mul- tiple time scale singular perturbation problems. SIAM J. Appl. Dyn. Syst. , 16(3):1425–1452, 2017

    work page 2017

  62. [62]

    Chen and R.E

    J. Chen and R.E. O’Malley, Jr. On the asymptotic solution of a two-parameter boundary value problem of chemical reactor theory. SIAM J. Appl. Math. , 26(4):717–729, 1974

    work page 1974

  63. [63]

    Cloez and M

    B. Cloez and M. Hairer. Exponential ergodicity for Markov processes with random switching. Bernoulli, 21:505–536, 2015

    work page 2015

  64. [64]

    Conforto and L

    F. Conforto and L. Desvillettes. Rigorous passage to the limit in a system of reaction–diffusion equations towards a system including cross diffusions. Commun. Math. Sci., 12(3):457–472, 2014

    work page 2014

  65. [65]

    Conforto, L

    F. Conforto, L. Desvillettes, and C. Soresina. About reaction–diffusion systems involving the Holling-type II and the Beddington–DeAngelis functional responses for predator–prey models. Nonlinear Differ. Equ. Appl. , 25(3):24, 2018

    work page 2018

  66. [66]

    S. F. Cooke and T. V. P. Bliss. Plasticity in the human central nervous system. Brain, 129(7):1659–1673, 2006

    work page 2006

  67. [67]

    Additive noise de- stroys a pitchfork bifurcation

    Hans Crauel and Franco Flandoli. Additive noise de- stroys a pitchfork bifurcation. J. Dynam. Differential Equations, 10(2):259–274, 1998

    work page 1998

  68. [68]

    On absolute continuity of invari- ant measures associated with a piecewise-deterministic Markov processes with random switching between flows

    Dawid Czapla, Katarzyna Horbacz, and Hanna Wo- jew´ odka-´Sciazko. On absolute continuity of invari- ant measures associated with a piecewise-deterministic Markov processes with random switching between flows. Available at https://arxiv.org/abs/2004.06798 , 2021

    work page arXiv 2004

  69. [69]

    Da Prato and J

    G. Da Prato and J. Zabczyk. Ergodicity for infinite- dimensional systems, volume 229 of London Mathemat- ical Society Lecture Note Series . Cambridge University Press, Cambridge, 1996

    work page 1996

  70. [70]

    Spike Timing-Dependent Plasticity of Neural Circuits.Neuron, 44(1):23–30, 2004

    Yang Dan and Mu-ming Poo. Spike Timing-Dependent Plasticity of Neural Circuits.Neuron, 44(1):23–30, 2004

    work page 2004

  71. [71]

    M.H.A. Davis. Piecewise-deterministic Markov pro- cesses: a general class of nondiffusion stochastic models. J. Roy. Statist. Soc. Ser. B , 46(3):353–388, 1984. With discussion

    work page 1984

  72. [72]

    Degn, L.F

    H. Degn, L.F. Olsen, and J.W. Perram. Bistability, os- cillation, and chaos in an enzyme reaction. Ann. N. Y. Acad. Sci., 316(1):623–637, 1979

    work page 1979

  73. [73]

    Desroches, J

    M. Desroches, J. Guckenheimer, C. Kuehn, B. Krauskopf, H. Osinga, and M. Wechselberger. Mixed-mode oscillations with multiple time scales. SIAM Rev., 54(2):211–288, 2012

    work page 2012

  74. [74]

    Desroches, B

    M. Desroches, B. Krauskopf, and H.M. Osinga. The geometry of mixed-mode oscillations in the Olsen model for the perioxidase-oxidase reaction. DCDS-S, 2(4):807– 827, 2009

    work page 2009

  75. [75]

    Desvillettes and C

    L. Desvillettes and C. Soresina. Non-triangular cross- diffusion systems with predator–prey reaction terms. Ric. Mat., 68(1):295–314, 2019

    work page 2019

  76. [76]

    Desvillettes and A

    L. Desvillettes and A. Trescases. New results for tri- angular reaction cross diffusion system. J. Math. Anal. Appl., 430(1):32–59, 2015

    work page 2015

  77. [77]

    DeVille, N

    L. DeVille, N. Sri Namachchivaya, and Z. Rapti. Sta- bility of a stochastic two-dimensional non-Hamiltonian system. SIAM J. Appl. Math. , 71(4):1458–1475, 2011

    work page 2011

  78. [78]

    di Bernardo, C.J

    M. di Bernardo, C.J. Budd, A.R. Champneys, and 30 P. Kowalczyk. Piecewise-smooth Dynamical Systems , volume 163 of Applied Mathematical Sciences. Springer, 2008

    work page 2008

  79. [79]

    The Morris-Lecar neuron model embeds a leaky integrate- and-fire model

    Susanne Ditlevsen and Priscilla Greenwood. The Morris-Lecar neuron model embeds a leaky integrate- and-fire model. Journal of Mathematical Biology , 67(2):239–259, 2013

    work page 2013

  80. [80]

    T.S. Doan, M. Engel, J.S.W. Lamb, and M. Rasmussen. Hopf bifurcation with additive noise. Nonlinearity, 31(10):4567–4601, 2018

    work page 2018

Showing first 80 references.