À propos de canards (Apropos canards)

Abstract

We extend canard theory of singularly perturbed systems to the general case of slow and fast dimensions, with and arbitrary. A folded critical manifold of a singularly perturbed system, a generic requirement for canards to exist, implies that there exists a local -dimensional center manifold spanned by the slow variables and the critical eigendirection of the fast variables. If one further assumes that the nonzero eigenvalues of the Jacobian matrix of the fast equation have all negative real part, then the -dimensional singularly perturbed problem is locally governed by the flow on the -dimensional center manifold. By using the blow-up technique (a desingularization procedure for folded singularities) we then show that the local flow near a folded singularity of a -dimensional folded critical manifold is, to leading order, governed by a three-dimensional canonical system for any . Consequently, results on generic canards from the well-known case can be extended to the general case .