Canard Cycles: from Birth to Transition

Peter de Maesschalck (1) , Freddy Dumortier (1) , Robert Roussarie (2)

1 Faculty of Sciences - Department of Mathematics and Statistics [Hasselt]
2 IMB - Institut de Mathématiques de Bourgogne [Dijon]

This book offers the first systematic account of canard cycles, an intriguing phenomenon in the study of ordinary differential equations. The canard cycles are treated in the general context of slow-fast families of two-dimensional vector fields. The central question of controlling the limit cycles is addressed in detail and strong results are presented with complete proofs. In particular, the book provides a detailed study of the structure of the transitions near the critical set of non-isolated singularities. This leads to precise results on the limit cycles and their bifurcations, including the so-called canard phenomenon and canard explosion. The book also provides a solid basis for the use of asymptotic techniques. It gives a clear understanding of notions like inner and outer solutions, describing their relation and precise structure. The first part of the book provides a thorough introduction to slow-fast systems, suitable for graduate students. The second and third parts will be of interest to both pure mathematicians working on theoretical questions such as Hilbert's 16th problem, as well as to a wide range of applied mathematicians looking for a detailed understanding of two-scale models found in electrical circuits, population dynamics, ecological models, cellular (FitzHugh–Nagumo) models, epidemiological models, chemical reactions, mechanical oscillators with friction, climate models, and many other models with tipping points.

Mots clés: Canard cycles; Slow-fast systems; limit cycles; vector field; relaxation oscillations; slow-fast bifurcations; singular pertubation theory; blow up method; transition asymptotics; finite cyclicity; Ordinary Differential Equations

Domaines: Mathématiques [math]

Peter de Maesschalck, Freddy Dumortier, Robert Roussarie. Canard Cycles: from Birth to Transition. Springer International Publishing, 73, 2021, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 978-3-030-79232-9. ⟨10.1007/978-3-030-79233-6⟩. ⟨hal-03476667⟩