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.
