Asymptotic study of planar canard solutions

Abstract

We are interested in the asymptotic study of canard solutions in real singularly perturbed first order ODE of the form $\varepsilon u'=\Psi(x,u,a,\varepsilon)$, where $\varepsilon>0$ is a small parameter, and $a\in\mathbb R$ is a real control parameter. An operator $\Xi_\eta$ was defined to prove the existence of canard solutions. This demonstration allows us to conjecture the existence of a generalized asymptotic expansion in fractional powers of $\varepsilon$ for those solutions. In this note, we propose an algorithm that computes such an asymptotic expansions for the canard solution. Furthermore, those asymptotic expansions remain uniformly valid.