Canard cycles : from birth to transition / Peter De Maesschalck ; Freddy Dumortier ; Robert Roussarie.

Format:

Book

Author/Creator:

Maesschalck, Peter de, author.

Dumortier, Freddy, author.

Roussarie, Robert H., author.

Series:

Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, Bd. 73.

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / Series of modern surveys in mathematics, 0071-1136 ; volume 73

Language:

English

Subjects (All):

Singular perturbations (Mathematics).

Vector fields.

Bifurcation theory.

Physical Description:

xxi, 408 pages : illustrations (some color) ; 25 cm.

Place of Publication:

Cham, Switzerland : Springer, [2021]

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.

Contents:

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.

Notes:

Includes bibliographical references (pages 401-405) and index.

Current copyright fee: GBP19.00 42\0.

ISBN:

3030792323

9783030792329

OCLC:

1252415084

Publisher Number:

10.1007/978-3-030-79233-6