| Titre : | Attractors and bifurcations of chaotic systems |
| Auteurs : | Faiza Zaamoune, Auteur ; Tidjani Menacer, Directeur de thèse |
| Type de document : | Thése doctorat |
| Editeur : | Biskra [Algérie] : Faculté des Sciences Exactes et des Sciences de la Nature et de la Vie, Université Mohamed Khider, 2022 |
| Format : | 1 vol. (89 p.) / couv. ill. en coul / 30 cm |
| Langues: | Français |
| Mots-clés: | Dynamical Systems, Chaos, Hidden attractors, Hidden bifurcation, modality of an odd number of spirals, Saturated function series, multi-spirals chaotic attractor, Symmetry. |
| Résumé : |
The hidden bifurcation idea was discovered by the core idea of the Leonov and Kuznetsov method for searching hidden attractors (i.e., homotopy and numerical continuation) differently in order to uncover hidden bifurcations governed by a homotopy parameter ɛ while keeping the numbers of spirals. This idea was first discovered by Menacer et al. In 2016, in the multispiral Chua system, The first part of this thesis is devoted to providing a basic understanding of dynamic systems and chaos, followed by an introduction to the hidden attractors, history, and definitions. An effective procedure for the numerical localization of hidden attractors in multidimensional dynamical systems has been presented by Leonov et Kuznetsov. In this part, we end with the study of hidden attractors in the Chua system. The second part of the analysis consists of first, hidden modalities of spirals of chaotic attractor via saturated function series and numerical results. Before reaching the asymptotic attractor which possesses an even number of spirals, these latter are generated one after one until they reach their maximum number, matching the value fixed by ɛ. Then, we end up by symmetries in hidden bifurcation routes to multiscroll chaotic attractors generated by saturated function series. The method to find such hidden bifurcation routes (HBR) depends upon two parameters. |
| Sommaire : |
Contents Dedicace i Thanks ii Table of Contents iii List of Figures vi Introduction 1 1 Dynamical Systems and Chaos 5 1.1 Introduction5 1.2 Important De…nitions and Notations 6 1.2.1 Phase Space .7 1.2.2 Conservative Systems and Dissipative Systems 8 1.2.3 The Poincare Map 8 1.2.4 Critical Points 8 1.2.5 Attractors of Dissipative Systems 10 1.3 Qualitative Study of Dynamic Systems 13 1.3.1 Linearization of Dynamic Systems 13 1.3.2 Concept of Stability 14 1.3.3 Hartmann-Grobman Theorem . 17 iii1.3.4 Central Manifold Theorem 17 2 Bifurcation Theory 22 2.1 Introduction 22 2.2 Bifurcations in Codimension 1 23 2.3 Chaos theory 30 2.3.1 Chaos Properties .31 2.3.2 Lyapunov’s Exponents32 2.3.3 Paths to Chaos 33 3 Hidden Attractors 35 3.1 Introduction . 35 3.2 Self-Excited Attractors 37 3.3 Hidden Oscillations 38 3.4 Analytical-Numerical Method for Hidden Attractor Localization 42 3.4.1 Example (Hidden Attractor for Chua’s System) . . . . . . . . . . . 45 4 Hidden Modalities of Spirals of Chaotic Attractor via Saturated Function Series and Numerical Results 50 4.1 Introduction 50 4.2 1-D n-Scroll Chaotic Attractors From Saturated Function Series 51 4.3 Recovering Hidden Bifurcation in a Multispiral Chaotic Attractor 55 4.3.1 Numerical Results of Hidden Bifurcations . . . . . . . . . . . . . . 57 4.3.2 The In‡uence of the Integration Duration Procedure for Unveiling Hidden Modalities of Odd Number of Spirals . . . . . . . . . . . . . 60 5 Symmetries in Hidden Bifurcation Routes to Multiscroll Chaotic Attractors Generated by Saturated Function Series 67 ivTable of Contents 5.1 Introduction 67 5.2 Models and Properties of Bifurcation Routes 68 5.2.1 Numerical Calculation of Two hidden Bifurcations Routes . . . . . 68 5.2.2 Maximal Attractor Range Extension and Coding Order of Spirals Appearance 70 5.3 Symmetries of the Hidden Bifurcation Routes 71 5.3.1 Basic Cell 72 5.3.2 Symmetries 72 Conclusion 79 Bibliography 81 Annexe : Program in MATLAB for Hidden Bifurcation Saturated Function Series |
| En ligne : | http://thesis.univ-biskra.dz/5983/1/Memoire%20Doctorant%20Zaamoune%20Faiza.pdf |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/134 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
