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Canard cycles = from birth to transi...

De Maesschalck, Peter.

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Canard cycles = from birth to transition /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Canard cycles/ by Peter De Maesschalck, Freddy Dumortier, Robert Roussarie.
Reminder of title:	from birth to transition /
Author:	De Maesschalck, Peter.
other author:	Dumortier, Freddy.
Published:	Cham :Springer International Publishing : : 2021.,
Description:	xxi, 408 p. :ill. (some col.), digital ;25 cm.
[NT 15003449]:	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.
Contained By:	Springer Nature eBook
Subject:	Singular perturbations (Mathematics) -
Online resource:	https://doi.org/10.1007/978-3-030-79233-6
ISBN:	9783030792336

Canard cycles = from birth to transition /
De Maesschalck, Peter.

Canard cyclesfrom birth to transition /[electronic resource] :by Peter De Maesschalck, Freddy Dumortier, Robert Roussarie. - Cham :Springer International Publishing :2021. - xxi, 408 p. :ill. (some col.), digital ;25 cm. - Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A series of modern surveys in mathematics,v.732197-5655 ;. - Ergebnisse der mathematik und ihrer grenzgebiete. 3. folge / A series of modern surveys in mathematics ;v.73..

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.

ISBN: 9783030792336

Standard No.: 10.1007/978-3-030-79233-6doiSubjects--Topical Terms:

897295
Singular perturbations (Mathematics)

LC Class. No.: QA372 / .D4 2021

Dewey Class. No.: 515.352

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100 1 $a De Maesschalck, Peter. $3 3514684
245 1 0 $a Canard cycles $h [electronic resource] : $b from birth to transition / $c by Peter De Maesschalck, Freddy Dumortier, Robert Roussarie.
260 $a Cham : $b Springer International Publishing : $b Imprint: Springer, $c 2021.
300 $a xxi, 408 p. : $b ill. (some col.), digital ; $c 25 cm.
490 1 $a Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A series of modern surveys in mathematics, $x 2197-5655 ; $v v.73
505 0 $a 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.
520 $a 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.
650 0 $a Singular perturbations (Mathematics) $3 897295
650 0 $a Vector fields. $3 705428
650 0 $a Bifurcation theory. $3 544229
650 1 4 $a Dynamical Systems and Ergodic Theory. $3 891276
650 2 4 $a Vibration, Dynamical Systems, Control. $3 893843
650 2 4 $a Analysis. $3 891106
650 2 4 $a Complexity. $3 893807
700 1 $a Dumortier, Freddy. $3 3514685
700 1 $a Roussarie, Robert. $3 726120
710 2 $a SpringerLink (Online service) $3 836513
773 0 $t Springer Nature eBook
830 0 $a Ergebnisse der mathematik und ihrer grenzgebiete. 3. folge / A series of modern surveys in mathematics ; $v v.73. $3 3514686
856 4 0 $u https://doi.org/10.1007/978-3-030-79233-6
950 $a Mathematics and Statistics (SpringerNature-11649)