Presses universitaires de Louvain onixsuitesupport@onixsuite.com 20260528T0341Z eng

COM.ONIXSUITE.9782930344751 03 01 Presses universitaires de Louvain 01 SKU 70934 02 293034475X 03 9782930344751 15 9782930344751 02 01 00 BC 01 9.45 in 02 6.30 in 03 1.24 in 08 12.59 oz 01 24 cm 02 16 cm 03 1.24 cm 08 357 gr BE 10 02 1 01 02 Thèses de l'Université catholique de Louvain (UCL) 03 Thèses de la Faculté des sciences 01 01 Numéro 1 A Study of heteroclinic orbits for a class of fourth order ordinary differential equations 1 A01 01 Onixsuite Contributor ID 5688 Denis Bonheure Bonheure, Denis Denis Bonheure Né le 30 Juillet 1977 à Braine-l'alleud. Licencié en Sciences mathématiques de la Faculté des Sciences de l'Université Libre de Bruxelles. Porteur du diplôme de DEA Inter-universitaire en Mathématiques délivré par la Faculté des Sciences de l'Université catholique de Louvain 1 01 eng 00 217 03 10 SCI000000 24 INTERNET Sciences exactes 24 INTERNET Analyse mathématique 24 INTERNET Mathématiques 29 2012 3051 SCIENCES FONDAMENTALES 93 P 01 06 03 00 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 02 00 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... 03 00 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. 02 00 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... 04 00 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 </dir></dir> 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 </dir></dir> Open Questions 26 About the Presentation 26 Chapter 1. The VariationalMethods and Heteroclinics for Second Order </dir></dir> 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 </dir></dir> 1.3. Reversible Hamiltonian Systems 43 1.4. Periodic Hamiltonian Systems: Heteroclinic Chains 50 1.5. Notes and Comments 56 </dir></dir> 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 </dir></dir> 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 </dir></dir> 2.4. Notes and Comments 87 </dir></dir> 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 </dir></dir> 3.4. Notes and Comments 114 </dir></dir> 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 </dir></dir> 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 </dir></dir></dir> 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 </dir></dir> 5.4. Notes and Comments 147 </dir></dir> Bibliography 149 List of Figures 155 Index 157 01 00 03 02 01 D502 02 188 03 124 06 178f5f07c878ecf1abd505463e41c6de 07 9151 https://pul.uclouvain.be/resources/titles/29303100880670/images/05d0abb9a864ae4981e933685b8b915c/THUMBNAIL/9782930344751.jpg 17 20160216T0947Z 15 00 04 02 Texte intégral en PDF à télécharger gratuitement 02 01 E107 07 1512014000 https://pul.uclouvain.be/resources/titles/29303100880670/extras/70934_sciences_bonheure_1002419.pdf 06 3052405007518 Presses universitaires de Louvain 01 01 Dilicom PULOUVAIN 06 3052405007518 Presses universitaires de Louvain Louvain-la-Neuve BE 04 01 20040101 11 20040101 01 WORLD 02 01 GCOI 29303100880670 WORLD 00 03 06 3012405004818 CIACO - DUC 01 2 20 1 02 00 02 02 STD 02 17.90 01 R 6.00 16.89 1.01 EUR BE FR 00 02 06 3019000200508 Librairie Wallonie-Bruxelles 33 www.librairiewb.com/ http://www.librairiewb.com/ 01 2 20 1 04 00 02 02 STD 02 17.90 01 R 5.50 16.97 0.93 EUR