Canard cycles : from birth to transition / Peter De Maesschalck, Freddy Dumortier, Robert Roussarie.
De Maesschalck, Peter, author.; Dumortier, Freddy, author.; Roussarie, Robert H., author.
2021
QA372 .D4 2021
Available Online

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Title Canard cycles : from birth to transition / Peter De Maesschalck, Freddy Dumortier, Robert Roussarie.
Author De Maesschalck, Peter, author.
ISBN 9783030792336 (electronic bk.)
3030792331 (electronic bk.)
9783030792329
3030792323
DOI https://doi.org/10.1007/978-3-030-79233-6
Published Cham : Springer, [2021]
Copyright ©2021
Language English
Description 1 online resource : illustrations (some color)
Item Number 10.1007/978-3-030-79233-6 doi
Call Number QA372 .D4 2021
Dewey Decimal Classification 515/.352
Summary 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.
Bibliography, etc. Note Includes bibliographical references and index.
Access Note Access limited to authorized users.
Source of Description Online resource; title from PDF title page (SpringerLink, viewed August 19, 2021).
Added Author Dumortier, Freddy, author.
Roussarie, Robert H., author.
Series Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, Bd. 73. 0071-1136
Available in Other Form Canard cycles.
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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.

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