Extending Geometric Singular Perturbation Theory to Nonhyperbolic Points---Fold and Canard Points in Two Dimensions

Krupa, Martin; Szmolyan, Peter

doi:10.1137/s0036141099360919

Cited by 481 publications

(817 citation statements)

References 17 publications

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“…Fix a section out 1 via (25) in {r 1 = 0}. Finally, by Taylor expanding (25) for ε 1 small, we find that the orbit + 1 is tangent to the v 1 -axis as + 1 → P 1 .…”

Section: Lemma 24 the Eigenvalues Of (22a)-(22c) Linearized Atmentioning

confidence: 92%

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The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

2007

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The Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with cut-off was introduced in (Brunet and Derrida 1997 Shift in the velocity of a front due to a cut-off Phys. Rev. E 56 2597-604) to model N-particle systems in which concentrations less than ε = 1/N are not attainable. It was conjectured that the cut-off function, which sets the reaction terms to zero if the concentration is below the small threshold ε, introduces a substantial shift in the propagation speed of the corresponding travelling waves. In this paper, we prove the conjecture of Brunet and Derrida, showing that the speed of propagation is given by c crit (ε) = 2 − π 2 /(ln ε) 2 + O((ln ε) −3 ), as ε → 0, for a large class of cut-off functions. Moreover, we extend this result to a more general family of scalar reaction-diffusion equations with cut-off. The main mathematical techniques used in our proof are the geometric singular perturbation theory and the blow-up method, which lead naturally to the identification of the reasons for the logarithmic dependence of c crit on ε as well as for the universality of the corresponding leading-order coefficient (π 2 ).

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“…Fix a section out 1 via (25) in {r 1 = 0}. Finally, by Taylor expanding (25) for ε 1 small, we find that the orbit + 1 is tangent to the v 1 -axis as + 1 → P 1 .…”

Section: Lemma 24 the Eigenvalues Of (22a)-(22c) Linearized Atmentioning

confidence: 92%

“…Fix a section out 1 via (25) in {r 1 = 0}. Finally, by Taylor expanding (25) for ε 1 small, we find that the orbit + 1 is tangent to the v 1 -axis as + 1 → P 1 . In the more generic case when ≡ 0, the orbit + 1 can be constructed and the corresponding value of α found, in a similar fashion.…”

Section: Lemma 24 the Eigenvalues Of (22a)-(22c) Linearized Atmentioning

confidence: 92%

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

2007

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“…In the neighborhood of this equilibrium, the distinction between fast and slow variables is lost, and the analysis is further complicated by the fact that the two fold lines of the cusp surface, which themselves are already not normally hyperbolic, meet at the cusp point. Nevertheless, the system dynamics may be analyzed using the method of geometric desingularization, also known as the blowup method [2,3,4,5,10,11]. Here, the origin is blown up into a hyper-sphere, and the induced equilibria are either hyperbolic or semi-hyperbolic.…”

Section: Analysis Of a Singularly Perturbed Cusp By Means Of Geometrimentioning

confidence: 99%

“…Setting ε = 0, we obtain the so-called layer problem [10], in which the systems is reduced to m ODEs for the fast variable y which depend on the slow variable x as a parameter. Constrained equations are equivalent to equations without a constraint at points where D y g is invertible.…”

Section: Introductionmentioning

confidence: 99%

“…A challenging problem is to extend the analysis of [14] to the context of (1) with 0 < ε ≪ 1. Such an extension has been carried out for all known singularities in the cases of problems with dimension k ≤ 2 [1,10,12], with the exception of the cusp singularity. In this article, the main result is a complete analysis of the cusp singularity Figure 1: The cusp singularity and the infinitely fast foliation for systems (1) with 0 < ε ≪ 1, and therefore we complete the study of generic singularities in dimensions k ≤ 2 for problems (1).…”

Section: Introductionmentioning

confidence: 99%

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Geometric Desingularization of a Cusp Singularity in Slow–Fast Systems with Applications to Zeeman’s Examples

2013

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The cusp singularity -a point at which two curves of fold points meet-is a prototypical example in Takens' classification of singularities in constrained equations, which also includes folds, folded saddles, folded nodes, among others. In this article, we study cusp singularities in singularly perturbed systems for sufficiently small values of the perturbation parameter, in the regime in which these systems exhibit fast and slow dynamics. Our main result is an analysis of the cusp point using the method of geometric desingularization, also known as the blow-up method, from the field of geometric singular perturbation theory. Our analysis of the cusp singularity was inspired by the nerve impulse example of Zeeman, and we also apply our main theorem to it. Finally, a brief review of geometric singular perturbation theory for the two elementary singularities from the Takens' classification occurring for the nerve impulse example -folds and folded saddles -is included to make this article self-contained.

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A note on kernel methods for multiscale systems with critical transitions

Hamzi

Kuehn

Mohamed

2018

Math Methods in App Sciences

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We study the maximum mean discrepancy (MMD) in the context of critical transitions modelled by fast-slow stochastic dynamical systems. We establish a new link between the dynamical theory of critical transitions with the statistical aspects of the MMD. In particular, we show that a formal approximation of the MMD near fast subsystem bifurcation points can be computed to leading order. This leading order approximation shows that the MMD depends intricately on the fast-slow systems parameters, which can influence the detection of potential early-warning signs before critical transitions. However, the MMD turns out to be an excellent binary classifier to detect the change-point location induced by the critical transition. We cross-validate our results by numerical simulations for a van der Pol-type model. KEYWORDSbifurcation, critical transition, kernel methods, maximum mean discrepancy, multiscale system, time series, tipping point• Change-point detection: In a time series generated by a fast-slow SODE, there could be many different types and sizes of drastic jumps. Hence, it would not only be useful to develop an automatic and generic classifiers, 10-12 when we actually observe a critical transition, but also to cross validate a classifier against explicit low-dimensional models.Math Meth Appl Sci. 2019;42:907-917. wileyonlinelibrary.com/journal/mma

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Extending Geometric Singular Perturbation Theory to Nonhyperbolic Points---Fold and Canard Points in Two Dimensions

Cited by 481 publications

References 17 publications

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

Geometric Desingularization of a Cusp Singularity in Slow–Fast Systems with Applications to Zeeman’s Examples

A note on kernel methods for multiscale systems with critical transitions

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