Pinching of canards and folded nodes – nonsmooth approximation of slow-fast dynamics

(2013). Pinching of canards and folded nodes: nonsmooth approximation of slow-fast dynamics. General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: Explore Bristol Research is a digital archive and the intention is that deposited content should not be removed. However, if you believe that this version of the work breaches copyright law please contact [email protected] and include the following information in your message: • Your contact details • Bibliographic details for the item, including a URL • An outline of the nature of the complaint On receipt of your message the Open Access Team will immediately investigate your claim, make an initial judgement of the validity of the claim and, where appropriate, withdraw the item in question from public view. Sudden changes in a dynamical system can be modelled by mixtures of slow and fast timescales, or by combining smooth change with sudden switching. In sets of ordinary differential equations, the former are modelled using singular perturbations, the latter using discontinuities. The relation between the two is not well understood, and here we develop a method called pinching, which approximates a singularly perturbed dynamical system by a discontinuous one, by making a discontinuous change of variables. We study pinching in the context of the canard phenomenon at a folded node. The folded node is a singularity associated with loss of normal hyperbolicity in slow-fast systems with (at least) two slow variables, and canards are special solutions that characterize the local dynamics. Pinching yields an approximation in terms of the twofold singularity of discontinuous (Filippov) systems, which arises generically in three or more dimensions, and remains a subject of interest in its own right. The purpose of this paper is to study the relation between discontinuities and singular perturbations in dynamical systems. We do this by forming a discontinuous model of a well-studied singular perturbation problem using the method of pinching introduced in [7]. A comparison of phase portraits in the two different kinds of system, particularly in the works [4, 21, 26] and [15, 18], reveals similarities in the dynamics associated with certain singularities. A rigorous explanation for this similarity has not been given, however. The singular limit of a slow-fast system in [26] indeed produces a system that is in some sense discontinuous – split into a slow timescale and …