Planar Canards with Transcritical Intersections
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Planar Canards with Transcritical Intersections P. De Maesschalck
Received: 22 August 2014 / Accepted: 2 October 2014 © Springer Science+Business Media Dordrecht 2014
Abstract We review essential techniques in the study of families of periodic orbits of slowfast systems in the plane. The techniques are demonstrated by treating orbits passing through unfoldings of transcritical intersections of curves of singular points in the most generic setting. We show that such transcritical intersections can generate canard type orbits. The stability of limit cycles of canard type containing that pass near transcritical intersections is examined by means of the slow divergence integral. Keywords Slow-fast systems · Slow divergence integral · Canards · Singular perturbations · Transcritical intersection
1 Introduction Planar slow-fast vector fields are smooth ε-families of vector fields Xε on the plane for which Xε has one or more curves of singular points. Identifying vector fields with the differential equations governing their integral curves, we write x˙ = f (x, y) + εf1 (x, y) + O ε 2 , Xε : (1) y˙ = g(x, y) + εg1 (x, y) + O ε 2 . The slow-fast property means that f and g have a common divisor: (f, g) = (ϕf , ϕg) where ϕ(x, y) is a multiplicative function that is zero along one or more curves in the plane, called critical curves. Along these critical curves, the vector field Xε is of O(ε)-magnitude, implying that near the curves the speed is small. This slow speed is in contrast with the “normal” O(1) speed found elsewhere. The focus in this paper is towards the study of periodic orbits of slow-fast type. These are ε-families of periodic orbits that spend some time in an O(ε)neighborhood of the critical curves, and some time away from them. We are in particular
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P. De Maesschalck ( ) Hasselt University, Martelarenlaan 42, 3500 Hasselt, Belgium e-mail: [email protected]
P. De Maesschalck
interested in deriving techniques to compute the stability (and multiplicity) of such slow-fast periodic orbits close to the singular limit ε → 0. The study of slow-fast dynamical systems arises from the observation that many biological, chemical and physical phenomena occur along multiple time scales: some observables can be seen to change on a fast time scale, whereas others change slowly. The ratio of the time scales, denoted by ε, is typically small, and the whole purpose of singular perturbation theory is to identify the small parameter ε, to study the model near the asymptotic limit ε = 0 and to apply perturbation theory to predict the behavior for nonzero ε. As ε is the ratio of two time scales, it is clear that in the limit ε = 0, the so-call slow variables no longer change dynamically but are in a fixed state. (In general, the slow variables are gathered in critical manifolds, e.g. critical curves in the planar case.) This is an essential property of a singular perturbation (in contrast to a regular perturbation). In this paper, we will not discuss applications, but we will instead discuss (part of)
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