04135nam 2200625 450 991049456060332120230427131853.03-030-79233-110.1007/978-3-030-79233-6(CKB)4100000011995863(DE-He213)978-3-030-79233-6(MiAaPQ)EBC6694563(Au-PeEL)EBL6694563(PPN)257351132(EXLCZ)99410000001199586320220425d2021 uy 0engurnn#008mamaatxtrdacontentcrdamediacrrdacarrierCanard cycles from birth to transition /Peter De Maesschalck, Freddy Dumortier, Robert Roussarie1st ed. 2021.Cham, Switzerland :Springer,[2021]©20211 online resource (XXI, 408 p. 101 illus., 42 illus. in color.)Ergebnisse der Mathematik und ihrer Grenzgebiete ;Band 733-030-79232-3 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.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.Ergebnisse der Mathematik und ihrer Grenzgebiete ;Band 73.Singular perturbations (Mathematics)Vector fieldsBifurcation theoryPertorbacions singulars (Matemàtica)thubCamps vectorialsthubTeoria de la bifurcacióthubLlibres electrònicsthubSingular perturbations (Mathematics)Vector fields.Bifurcation theory.Pertorbacions singulars (Matemàtica)Camps vectorialsTeoria de la bifurcació515.392Maesschalck Peter De1222486Dumortier FreddyRoussarie Robert H.MiAaPQMiAaPQMiAaPQBOOK9910494560603321Canard cycles2835266UNINA