Canard Cycles : From Birth to Transition 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 Hilberts 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. Part I Basic Notions 1 Basic Definitions and Notions 2 Local Invariants and Normal Forms 3 The Slow Vector Field 4 Slow Fast Cycles 5 The Slow Divergence Integral 6 Breaking Mechanisms 7 Overview of Known Results Part II Technical Tools 8 Blow Up of Contact Points 9 Center Manifolds 10 Normal Forms 11 Smooth Functions on Admissible Monomials and More 12 Local Transition Maps Part III Results and Open Problems 13 Ordinary Canard Cycles 14 Transitory Canard Cycles with Slow Fast Passage Through a Jump Point 15 Transitory Canard Cycles with Fast Fast Passage Through a Jump Point 16 Outlook and Open Problems Index References.