A qualitative study of some classes of differential systems beyond the quadratic ones / Meryem Belattar  

Public ISBD Titre : A qualitative study of some classes of differential systems beyond the quadratic ones Type de document : document électronique Auteurs : Meryem Belattar, Auteur ; Rachid Cheurfa, Directeur de thèse Editeur : Setif:UFA Année de publication : 2024 Importance : 1 vol (78 f.) Format : 29 cm Langues : Français (fre) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Algebraic limit cycle  autonomous first integral  invariant curve  non-algebraic limit cycle  phase portrait  Poincaré disk  solvable system Index. décimale : 510 Mathématique Résumé : The qualitative theory of ordinary differential systems represents an important tool for identifying properties of solutions without the need for the explicit resolution of these systems. Our main objective in this thesis is to investigate and solve certain problems of the qualitative theory for three classes of planar autonomous nonlinear differential symmetric systems with real parameters. These classes are: (I) a third-degree system, (II) a ninth-degree oscillator system, and (III) a sixthdegree system. Using classical results and methods as well as mathematical tools from the qualitative theory of ordinary differential equations, we aim to address issues related to integrability, solvability, limit cycles and the classification of topological phase portraits in the Poincaré disk for these systems. Note de contenu : Contents List of Figures iii List of Tables v Notations vi Scientific production vii General introduction 1 I Fundamental concepts of planar differential systems 5 1 Background 6 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Continuous-time planar differential systems . . . . . . . . . . . . . . . . . . 7 1.3 First integrals and invariant curves . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Limit cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.1 Some results about limit cycles . . . . . . . . . . . . . . . . . . . . 12 1.4.2 Stability of limit cycles . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.3 Approaches for analyzing stability . . . . . . . . . . . . . . . . . . . 14 1.5 Equilibrium points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.1 Stability of equilibrium points . . . . . . . . . . . . . . . . . . . . . 16 1.5.2 Stability analysis of equilibrium points . . . . . . . . . . . . . . . . 17 1.6 Linearization and classification of equilibrium points . . . . . . . . . . . . . 18 1.7 Phase portraits in the plane . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.7.1 Phase portraits at simple equilibrium points . . . . . . . . . . . . . 21 1.7.2 Phase portraits at nonsimple equilibrium points . . . . . . . . . . . 28 1.8 Blow-up technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.8.1 Polar blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.8.2 Directional blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.9 Poincaré compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.9.1 Phase portrait in the Poincaré disk . . . . . . . . . . . . . . . . . . 35 1.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 II Contributions 37 2 A cubic planar system with non-algebraic limit cycles enclosing a focus 38 2.1 Introduction and the main result . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 Study of equilibrium points . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2.1 Finite equilibrium points . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2.2 Infinite equilibrium points . . . . . . . . . . . . . . . . . . . . . . . 42 2.3 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.3.1 The non-existence of limit cycles . . . . . . . . . . . . . . . . . . . 47 2.3.2 First integral and non-algebraic limit cycles . . . . . . . . . . . . . 48 2.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3 Nonlinear oscillators with first integrals and algebraic limit cycles 55 3.1 Introduction and the main result . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 The solutions of the quartic algebraic equations . . . . . . . . . . . . . . . 57 3.3 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3.1 First integral and algebraic limit cycles . . . . . . . . . . . . . . . . 58 3.3.2 Phase portraits of the vector field Y . . . . . . . . . . . . . . . . . 60 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4 An exactly solvable planar system of degree six with an explicit limit cycle 63 4.1 Introduction and the main result . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2.1 First integrals and solvability of Z . . . . . . . . . . . . . . . . . . . 65 4.2.2 Algebraic limit cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2.3 Phase portraits of the vector field Z and level curves . . . . . . . . 68 4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Conclusion and perspectives 70 Appendix 72 A Obtaining the exact solutions 72 Bibliography 74 Côte titre : DM/0199 En ligne : https://drive.google.com/file/d/18LmMT3WYvlZNQu6EY8zRWfTnCCgws9pr/view?usp=shari [...] Format de la ressource électronique : pdf A qualitative study of some classes of differential systems beyond the quadratic ones [document électronique] / Meryem Belattar, Auteur ; Rachid Cheurfa, Directeur de thèse . - [S.l.] : Setif:UFA, 2024 . - 1 vol (78 f.) ; 29 cm. Langues : Français (fre) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Algebraic limit cycle  autonomous first integral  invariant curve  non-algebraic limit cycle  phase portrait  Poincaré disk  solvable system Index. décimale : 510 Mathématique Résumé : The qualitative theory of ordinary differential systems represents an important tool for identifying properties of solutions without the need for the explicit resolution of these systems. Our main objective in this thesis is to investigate and solve certain problems of the qualitative theory for three classes of planar autonomous nonlinear differential symmetric systems with real parameters. These classes are: (I) a third-degree system, (II) a ninth-degree oscillator system, and (III) a sixthdegree system. Using classical results and methods as well as mathematical tools from the qualitative theory of ordinary differential equations, we aim to address issues related to integrability, solvability, limit cycles and the classification of topological phase portraits in the Poincaré disk for these systems. Note de contenu : Contents List of Figures iii List of Tables v Notations vi Scientific production vii General introduction 1 I Fundamental concepts of planar differential systems 5 1 Background 6 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Continuous-time planar differential systems . . . . . . . . . . . . . . . . . . 7 1.3 First integrals and invariant curves . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Limit cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.1 Some results about limit cycles . . . . . . . . . . . . . . . . . . . . 12 1.4.2 Stability of limit cycles . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.3 Approaches for analyzing stability . . . . . . . . . . . . . . . . . . . 14 1.5 Equilibrium points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.1 Stability of equilibrium points . . . . . . . . . . . . . . . . . . . . . 16 1.5.2 Stability analysis of equilibrium points . . . . . . . . . . . . . . . . 17 1.6 Linearization and classification of equilibrium points . . . . . . . . . . . . . 18 1.7 Phase portraits in the plane . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.7.1 Phase portraits at simple equilibrium points . . . . . . . . . . . . . 21 1.7.2 Phase portraits at nonsimple equilibrium points . . . . . . . . . . . 28 1.8 Blow-up technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.8.1 Polar blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.8.2 Directional blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.9 Poincaré compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.9.1 Phase portrait in the Poincaré disk . . . . . . . . . . . . . . . . . . 35 1.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 II Contributions 37 2 A cubic planar system with non-algebraic limit cycles enclosing a focus 38 2.1 Introduction and the main result . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 Study of equilibrium points . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2.1 Finite equilibrium points . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2.2 Infinite equilibrium points . . . . . . . . . . . . . . . . . . . . . . . 42 2.3 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.3.1 The non-existence of limit cycles . . . . . . . . . . . . . . . . . . . 47 2.3.2 First integral and non-algebraic limit cycles . . . . . . . . . . . . . 48 2.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3 Nonlinear oscillators with first integrals and algebraic limit cycles 55 3.1 Introduction and the main result . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 The solutions of the quartic algebraic equations . . . . . . . . . . . . . . . 57 3.3 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3.1 First integral and algebraic limit cycles . . . . . . . . . . . . . . . . 58 3.3.2 Phase portraits of the vector field Y . . . . . . . . . . . . . . . . . 60 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4 An exactly solvable planar system of degree six with an explicit limit cycle 63 4.1 Introduction and the main result . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2.1 First integrals and solvability of Z . . . . . . . . . . . . . . . . . . . 65 4.2.2 Algebraic limit cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2.3 Phase portraits of the vector field Z and level curves . . . . . . . . 68 4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Conclusion and perspectives 70 Appendix 72 A Obtaining the exact solutions 72 Bibliography 74 Côte titre : DM/0199 En ligne : https://drive.google.com/file/d/18LmMT3WYvlZNQu6EY8zRWfTnCCgws9pr/view?usp=shari [...] Format de la ressource électronique : pdf

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Code-barres	Cote	Support	Localisation	Section	Disponibilité
DM/0199	DM/0199	Thèse	Bibliothéque des sciences	Anglais	Disponible Disponible

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