Powered by OpenAIRE graph
Found an issue? Give us feedback
image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/ Journal of Different...arrow_drop_down
image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
Journal of Differential Equations
Article
License: Elsevier Non-Commercial
Data sources: UnpayWall
image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
Journal of Differential Equations
Article . 2016 . Peer-reviewed
License: Elsevier Non-Commercial
Data sources: Crossref
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
zbMATH Open
Article . 2016
Data sources: zbMATH Open
versions View all 3 versions
addClaim

Summability of canard-heteroclinic saddle connections

Authors: Karel Kenens;

Summability of canard-heteroclinic saddle connections

Abstract

In this article Gevrey properties of analytic invariant curves of analytic slow-fast systems \[ \begin{aligned} \dot{x} & =\varepsilon x, \\ \dot{y} & = \varphi(x)y+\varepsilon H(x,y,\varepsilon) \end{aligned} \] with \(\varphi(0)<0\) are studied. The formal invariant manifold is locally given as a series \(y=\sum_{n=0}^\infty y_n(x)\varepsilon^n\) that is Gevrey-1 in \(\varepsilon\). Using that the Borel transform of this formal solution coincides in some sector with an exact solution of at most exponential growth suffices to prove the 1-summability with respect to \(\varepsilon\) in the positive real direction. Based on this result the author then studies canard solutions that are generated if the two invariant manifolds from two different slow-fast saddle points are connected near a turning point of the slow manifold. The main theorem states that for a real analytic slow-fast system a heteroclinic saddle connection between two slow-fast saddle equilibria on the slow manifold is summable with respect to the singular parameter in the positive real direction uniformly for \(x\) in compact subsets of the domain of the saddle connection which do not include the turning point.

Country
Belgium
Related Organizations
Keywords

formal solution, Gevrey series, Gevrey series; Borel summation; Slow-fast systems; Singular perturbations; Canards, Borel transform, Canard solutions to ordinary differential equations, Gevrey series; Borel summation; slow-fast systems; singular perturbations; canards, Perturbations, asymptotics of solutions to ordinary differential equations, slow-fast systems, Borel summation, Singular perturbations for ordinary differential equations, canards, singular perturbations, Asymptotic expansions of solutions to ordinary differential equations, Invariant manifolds for ordinary differential equations

  • BIP!
    Impact byBIP!
    selected citations
    These citations are derived from selected sources.
    This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    1
    popularity
    This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
    Average
    influence
    This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    Average
    impulse
    This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
    Average
Powered by OpenAIRE graph
Found an issue? Give us feedback
selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
1
Average
Average
Average
Green
hybrid