Asymptotic study of planar canard solutions (Q2519139)

The author studies singularly perturbed first-order ordinary differential equations of the form \[ \varepsilon y' =\Psi (x,y,a,\varepsilon), \] where \(\varepsilon\) is a small positive parameter, and \(a\in \mathbb R\) is a real control parameter, \(y\) is a real function. The author defines some operator in order to prove the existence of canard solutions to the equation. Several cases for the operator are considered and the so-called combined asymptotic expansions method is used. These allow to conjecture the existence of a generalized asymptotic expansion in fractional powers of \(\varepsilon\) for those solutions. The author proposes an algorithm that computes such asymptotic expansions for the canard solution. Furthermore, asymptotic expansions remain uniformly valid.