Advances in Differential Equations

Canards and bifurcation delays of spatially homogeneous and inhomogeneous types in reaction-diffusion equations

Peter De Maesschalck, Tasso J. Kaper, and Nikola Popović

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Abstract

In ordinary differential equations of singular perturbation type, the dynamics of solutions near saddle-node bifurcations of equilibria are rich. Canard solutions can arise, which, after spending time near an attracting equilibrium, stay near a repelling branch of equilibria for long intervals of time before finally returning to a neighborhood of the attracting equilibrium (or of another attracting state). As a result, canard solutions exhibit bifurcation delay. In this article, we analyze some linear and nonlinear reaction-diffusion equations of singular perturbation type, showing that solutions of these systems also exhibit bifurcation delay and are, hence, canards. Moreover, it is shown for both the linear and the nonlinear equations that the exit time may be either spatially homogeneous or spatially inhomogeneous, depending on the magnitude of the diffusivity.

Article information

Source
Adv. Differential Equations Volume 14, Number 9/10 (2009), 943-962.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355863335

Mathematical Reviews number (MathSciNet)
MR2548283

Zentralblatt MATH identifier
1184.35042

Subjects
Primary: 35K57: Reaction-diffusion equations 34E20: Singular perturbations, turning point theory, WKB methods 37G10: Bifurcations of singular points

Citation

De Maesschalck, Peter; Popović, Nikola; Kaper, Tasso J. Canards and bifurcation delays of spatially homogeneous and inhomogeneous types in reaction-diffusion equations. Adv. Differential Equations 14 (2009), no. 9/10, 943--962. https://projecteuclid.org/euclid.ade/1355863335.


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