A study of heteroclinic orbits for a class of fourth order ordinary differential equations

In qualitative theory of differential equations, an important role is played by special classes of solutions, like periodic solutions or solutions to some boundary value problems. When a system of ordinary differential equations has equilibria, i.e... Continuer

In qualitative theory of differential equations, an important role is played by special classes of solutions, like periodic solutions or solutions to some boundary value problems. When a system of ordinary differential equations has equilibria, i.e. co


In qualitative theory of differential equations, an important role is played by special classes of solutions, like periodic solutions or solutions to some boundary value problems. When a system of ordinary differential equations has equilibria, i.e. constant solutions, whose stability properties are known, it is significant to search for connections between them by trajectories of solutions of the given system. These are called homoclinic or heteroclinic, according to whether they describe a loop based at one single equilibrium or they "start" and "end" at two distinct equilibria. This thesis is devoted to the study of heteroclinic solutions for a specific class of ordinary differential equations related to the Extended Fisher-Kolmogorov equation and the Swift-Hohenberg equation. These are semilinear fourth order bi-stable evolution equations which appear as mathematical models for problems arising in Mechanics, Chemistry and Biology. For such equations, the set of bounded stationary solutions is of great interest. These solve an autonomous fourth order equation. In this thesis, we focus on such equations having a variational structure. In that case, the solutions are critical points of an associated action functional defined in convenient functional spaces. We then look for heteroclinic solutions as minimizers of the action functional. Our main contributions concern existence and multiplicity results of such global and local minimizers in the case where the functional is defined from sign changing Lagrangians. The underlying idea is to impose conditions which imply a lower bound on the action over all admissible functions. We then combine classical arguments of the Calculus of Variations with careful estimates on minimizing sequences to prove the existence of a minimum.


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Spécifications


Éditeur
Presses universitaires de Louvain
Partie du titre
Numéro 1
Auteur
Denis Bonheure,
Collection
Thèses de la Faculté des sciences
Langue
anglais
Catégorie (éditeur)
Sciences exactes > Mathématiques > Analyse mathématique
Catégorie (éditeur)
Sciences exactes > Mathématiques
Catégorie (éditeur)
Sciences exactes
BISAC Subject Heading
SCI000000 SCIENCE
Code publique Onix
06 Professional and scholarly
CLIL (Version 2013 )
3051 SCIENCES PURES
Date de première publication du titre
2004
Subject Scheme Identifier Code
Thema subject category: P
Type d'ouvrage
Thèse

Livre broché


Details de produit
2 A4
Date de publication
2004
ISBN-13
978-2-93034-475-1
Ampleur
Nombre de pages de contenu principal : 217
Code interne
70934
Format
16 x 24 x 1,2 cm
Poids
357 grammes
Prix
17,90 €
ONIX XML
Version 2.1, Version 3

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Sommaire


Préface vii

Avant-propos xi

Liste des publications lix

A Study of Heteroclinic Orbits for a Class of Fourth Order Ordinary Differential Equations 1

Contents 3

Introduction 5

An overview 7

Methods 7

The Model Equation 8

The Variational Approach 12

General Framework 14

Outline of the Thesis 16

Minimization Methods and Second Order Systems 16

Minimization of Positive Functionals 17

Sign Changing Lagrangians 20

Multi-transition Heteroclinics 22

Connections between Non-consecutive Equilibria 24

Open Questions 26

About the Presentation 26

Chapter 1. The VariationalMethods and Heteroclinics for Second Order

Equations and Systems 27

1.1. The Variational Methods 27

1.2. Basic Arguments for Scalar Equations 33

1.2.1. Phase Plane Analysis 33

1.2.2. The Autonomous Case 34

1.2.3. The Periodic Case 38

1.2.4. The Bounded Case 41

1.3. Reversible Hamiltonian Systems 43

1.4. Periodic Hamiltonian Systems: Heteroclinic Chains 50

1.5. Notes and Comments 56

Chapter 2. Minimization of Positive Functionals 61

2.1. The Extended Fisher-Kolmogorov Equation 62

2.2. Double-well Potentials with Degenerate Minima 67

2.2.1. Proof of Proposition 2.9 73

2.3. Qualitative Properties of the Minimizers 78

2.3.1. Clipping 79

2.3.2. Monotonicity of the Transitions 81

2.3.3. Oscillations in the Tails 83

2.3.4. Symmetric Functionals in the Saddle-foci Case 87

2.4. Notes and Comments 87

Chapter 3. Sign Changing Lagrangians 91

3.1. Functionals with Sign Changing Acceleration Coefficient 92

3.2. Functionals of Swift-Hohenberg Type 97

3.3. Non-symmetric Functionals 103

3.3.1. Analysis of the Local Minimizers Close to a Saddle-focus Equilibrium 104

3.3.2. Existence of a Minimizer 107

3.4. Notes and Comments 114

Chapter 4. Multi-transition Connections 117

4.1. Homotopy Classes of Heteroclinic Solutions 117

4.2. Multi-transition Heteroclinics 119

4.2.1. Functionals of Swift-Hohenberg Type 120

4.2.2. Sign Changing Acceleration Coefficient 132

4.3. Multi-transition Homoclinics 135

4.4. Notes and Comments 137

Chapter 5. A Ginzburg-Landau Model for Ternary Mixtures:

Connections Between Non-consecutive Equilibria 141

5.1. A Ginzburg-Landau Model for Ternary Mixtures 141

5.1.1. Binary Fluids 141

5.1.2. Ternary Mixtures 142

5.2. The Fourth Order Model 143

5.4. Notes and Comments 147

Bibliography 149

List of Figures 155

Index 157


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