LEADER 04135nam 2200625 450 001 9910494560603321 005 20230427131853.0 010 $a3-030-79233-1 024 7 $a10.1007/978-3-030-79233-6 035 $a(CKB)4100000011995863 035 $a(DE-He213)978-3-030-79233-6 035 $a(MiAaPQ)EBC6694563 035 $a(Au-PeEL)EBL6694563 035 $a(PPN)257351132 035 $a(EXLCZ)994100000011995863 100 $a20220425d2021 uy 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aCanard cycles $efrom birth to transition /$fPeter De Maesschalck, Freddy Dumortier, Robert Roussarie 205 $a1st ed. 2021. 210 1$aCham, Switzerland :$cSpringer,$d[2021] 210 4$dŠ2021 215 $a1 online resource (XXI, 408 p. 101 illus., 42 illus. in color.) 225 1 $aErgebnisse der Mathematik und ihrer Grenzgebiete ;$vBand 73 311 $a3-030-79232-3 327 $aPart 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. 330 $aThis book offers the first systematic account of canard cycles, an intriguing phenomenon in the study of ordinary differential equations. The canard cycles are treated in the general context of slow-fast families of two-dimensional vector fields. The central question of controlling the limit cycles is addressed in detail and strong results are presented with complete proofs. In particular, the book provides a detailed study of the structure of the transitions near the critical set of non-isolated singularities. This leads to precise results on the limit cycles and their bifurcations, including the so-called canard phenomenon and canard explosion. The book also provides a solid basis for the use of asymptotic techniques. It gives a clear understanding of notions like inner and outer solutions, describing their relation and precise structure. The first part of the book provides a thorough introduction to slow-fast systems, suitable for graduate students. The second and third parts will be of interest to both pure mathematicians working on theoretical questions such as Hilbert's 16th problem, as well as to a wide range of applied mathematicians looking for a detailed understanding of two-scale models found in electrical circuits, population dynamics, ecological models, cellular (FitzHugh?Nagumo) models, epidemiological models, chemical reactions, mechanical oscillators with friction, climate models, and many other models with tipping points. 410 0$aErgebnisse der Mathematik und ihrer Grenzgebiete ;$vBand 73. 606 $aSingular perturbations (Mathematics) 606 $aVector fields 606 $aBifurcation theory 606 $aPertorbacions singulars (Matemātica)$2thub 606 $aCamps vectorials$2thub 606 $aTeoria de la bifurcaciķ$2thub 608 $aLlibres electrōnics$2thub 615 0$aSingular perturbations (Mathematics) 615 0$aVector fields. 615 0$aBifurcation theory. 615 7$aPertorbacions singulars (Matemātica) 615 7$aCamps vectorials 615 7$aTeoria de la bifurcaciķ 676 $a515.392 700 $aMaesschalck$b Peter De$01222486 702 $aDumortier$b Freddy 702 $aRoussarie$b Robert H. 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910494560603321 996 $aCanard cycles$92835266 997 $aUNINA