Techniques in the Theory of Local Bifurcations: Blow-Up, Normal Forms, Nilpotent Bifurcations, Singular Perturbations

Dumortier, Freddy

doi:10.1007/978-94-015-8238-4_2

Cited by 131 publications

(151 citation statements)

References 15 publications

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“…To the best of our knowledge, the blow-up technique was first used in studying limit cycles near a cuspidal loop in [15]. The method has since been successfully applied, including in [16], as an extension of the more classical geometric singular perturbation theory to problems in which normal hyperbolicity is lost; see also [13,14,17,18,25,26,30] and the references therein.…”

Section: Theorem 11 For Any Reaction-diffusion Equation Of the Formmentioning

confidence: 99%

“…Here, we note that 2 is the unique orbit that is asymptotic to Q + 2 in K 2 when ≡ 0. Moreover, we define the point P in 2 = 2 ∩ in 2 as the intersection of 2 with the section in 2 ; more precisely, P in 2 = (1, −2, 0) by (18).…”

Section: Dynamics In the Rescaling Chart Kmentioning

confidence: 99%

“…Moreover, recall the expressions (18) and (25) For the connection between Q − 1 and P 1 , recall that in the invariant plane {ε 1 = 0}, system (22a)-(22c) corresponds precisely to the original FKPP equation (1). Hence, the heteroclinic orbit connecting Q − 1 and P 1 is given by the corresponding FKPP orbit, denoted by − 1 above, after blow-up.…”

Section: Proposition 27 Let ϕ Be a Cut-off Function Which Satisfiesmentioning

confidence: 99%

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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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Section: Theorem 11 For Any Reaction-diffusion Equation Of the Formmentioning

confidence: 99%

Section: Dynamics In the Rescaling Chart Kmentioning

confidence: 99%

Section: Proposition 27 Let ϕ Be a Cut-off Function Which Satisfiesmentioning

confidence: 99%

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

2007

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“…By contrast, points where D y g is not invertible are called singularities. The set S 0 = {g(x, y) = 0} is referred to as the constraining manifold in [14]; it is the phase space of (2). In the literature on singular perturbation theory, S 0 is also known as the critical manifold [8], or slow manifold.…”

Section: Introductionmentioning

confidence: 99%

“…A point away from S 0 moves infinitely fast along a stable fast fiber, following the dynamics of (3) with ε = 0, until it reaches a stable branch of S 0 . On S 0 , the dynamics switches to (2). If the corresponding solution reaches a singularity or a bifurcation point (loss of stability of S 0 ), then the dynamics switches back to (3).…”

Section: Introductionmentioning

confidence: 99%

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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1995

Applied Nonlinear Dynamics

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Techniques in the Theory of Local Bifurcations: Blow-Up, Normal Forms, Nilpotent Bifurcations, Singular Perturbations

Cited by 131 publications

References 15 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

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