Etude d'equations differentielles singulierement perturbees au voisinage d'un point tournant
Abstract
Description
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 $\{θ_1< \arg(x)<θ_2, and |x|<|X_l| |ε|^{1/(p+1)} \}$. If we introduce thereafter a p-parameter $α$ in the equation, we have (for some particular values of the parameter, depending from $ε$) 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 $ε$ for the values of $α$ corresponding to canards solutions.
110 pages, Ph-D thesis, in french
110 pages, Ph-D thesis, in french