Asymptotic behaviour of solutions near a turning point: The example of the Brusselator equation

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Abstract

The Brusselator equation is an example of a singularly perturbed differential equation with an additional parameter. It has two turning points: at x = 0 and x = -1. We study some properties of so-called canard solutions, that remain bounded in a full neighbourhood of 0 and in the largest possible domain; the main goal is the complete asymptotic expansion of the difference between two values of the additional parameter corresponding to such solutions. For this purpose we need a study of behaviour of the solutions near a turning point; here we prove that, for a large class of equations, if 0 is a turning point of order p, any solution y not exponentially large has, in some sector centred at 0, an asymptotic behaviour (when ε → 0) of the form ∑ Yn(x/ε′) ε′n, where ε′p+1 = ε, for x = ε′X with X large enough, but independent of ε. In the Brusselator case, we moreover compute a Stokes constant for a particular nonlinear differential equation. © 2005 Elsevier Inc. All rights reserved.

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APA

Matzinger, É. (2006). Asymptotic behaviour of solutions near a turning point: The example of the Brusselator equation. Journal of Differential Equations, 220(2), 478–510. https://doi.org/10.1016/j.jde.2005.06.028

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