Etude d'equations differentielles singulierement perturbees au voisinage d'un point tournant

Abstract

Consider a singularly perturbed ordinary differential equation, admitting 0 as turning point of order p. We study the behaviour, in the complex plane, of the solutions of this equation in the neighborhood of 0. We prove that the domain of these solutions contains sectors like $\{\theta_1< \arg(x)<\theta_2, and |x|<|X_l| |\epsilon|^{1/(p+1)} \}$. If we introduce thereafter a p-parameter $\alpha$ in the equation, we have (for some particular values of the parameter, depending from $\epsilon$) canards solutions, id est solutions which are bounded in a whole neighborhood of the turning point. These two results are used for two examples, one of them is the Van der Pol equation; we can then find for these two equations an equivalent of the coefficients of the asymptotic serie in $\epsilon$ for the values of $\alpha$ corresponding to canards solutions.