| Titre : | Stability and Bifurcations and Control in Fractional Order Chaotic Systems |
| Auteurs : | Besma Chettouh, 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, 2024 |
| Format : | 1 vol. (84 p.) / couv. ill. en coul / 30 cm |
| Langues: | Anglais |
| Mots-clés: | Fractional order, Dynamic system, Stability, Bifurcations, Hopf bifurcation, Chaos, Control, E¤ect of fractional order, Jerk system, Localisation. |
| Résumé : |
The inclusion of fractional-order dynamics in the study of nonlinear systems has broadened our understanding of complex behaviors, such as stability, chaos and bifurcations, and has opened up new possibilities in control theory. These systems involve derivatives and integrals of non-integer order, introducing a new level of flexibility and versatility in modeling realworld phenomena. This thesis aims to study the stability and bifurcations in a fractional order chaotic systems and the control of chaos. To achieve our goal we introduced in the first tow chapters the necessary basic notions such as: fractional derivation, chaos theory, stability of fractional systems and bifurcation theory. The main results of this thesis are presented in the last tow chapters where we gave the necessary and su¢ cient conditions for stability, we showed the existence of Hopf bifurcations in both cases: integer and fractional also we proved the effect of fractional order in the critical point location of Hopf’s bifurcation points, stability and chaos control. |
| Sommaire : |
Table of figures viii Introduction 1 I Preliminaries 5 1 Stability of Fractional Dynamical Systems 6 1.1 Fractional Differentiation . . . . . . . . . . . . . 6 1.1.1 Grunwald-Letnikov Fractional Derivative . . . . . . . . 6 1.1.2 Riemann-Liouville Fractional Derivative . . . . . . . 8 1.1.3 Caputo Fractional Derivative . . . . . . . 1.1.4 Laplace Transforms of Fractional Derivatives . . . . .. 11 1.1.5 Comparison Between Caputo and Riemann-Liouville Derivative’s . . 13 1.2 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.1 Continuous Dynamical Systems . . . . . . 14 1.2.2 Discrete Dynamical Systems . . . . . . . . . . . . . 15 1.2.3 Autonomous and Non-autonomous Systems . . . . . . 15 1.2.4 Poincaré Section . . . . . . . . . . . . . . . . . . . 15 1.3 Fractional Di¤erential Equations (FDE) . .. . . . 16 1.3.1 Cauchy Problem . . . . . . . . . . .. . 16 v1.3.2 Existence and Uniqueness 1.3.3 Numerical Solving Fractional Equations . . . . .. 18 1.4 Stability of Fractional Dynamical System . . . . . . . . . . . 19 1.4.1 Equilibrium Point . . . . . . . . . . . . . . . . . . . 20 1.4.2 Stability of Autonomous Linear Systems . . . . . . . . . 21 1.4.3 Stability of Nonlinear Systems (Linearization) . . . . . . . 23 1.4.4 Hartman-Grobman Theorem . . . . . . . . . . . . . .. 24 1.4.5 Fractional Order Routh-Hurwitz Criterion . . . . . . . . . . 25 1.4.6 Fractional-Order Extension of Lyapunov Direct Method . . . . . . . . 27 2 Chaos and Bifurcations Theory 30 2.1 Basic tools . . . . . . . . . . . . . . . . . 30 2.1.1 Lyapunov Exponents . . . . . . . . .. . . . 30 2.1.2 Attractors and Basin of Attraction . . . . . . . . 34 2.2 Chaos Theory . . . . . . . . . 37 2.2.1 Characteristics of Chaos . . . . . . . . . . . 37 2.2.2 Some Applications of Chaos . . . . . . . . .. . 40 2.3 Bifurcation Theory . . . . . . . . . . . . . . . 41 2.3.1 Saddle-Node Bifurcation . . . . . . . . . . . 42 2.3.2 Transcritical Bifurcation . . . . . . . . . . . . 44 2.3.3 Pitchfork Bifurcation . . . . . . . . . . . . 46 2.3.4 Hopf Bifurcation . . . . . . . . . . . . . . . . . 50 II Main Results 54 3 Fractional order effect on the localisation of Hopf bifurcation point 55 vi 3.1 Description and Stability Analysis of the Model . . . . . . . . 56 3.2 Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . 59 3.2.1 Integer Order Case . . . . . . . . . . . 59 3.2.2 Fractional Order Case . . . . . . . . . . . . . . . . . . . 61 3.3 Numerical Results . . . . . . . . . . . . . 65 3.3.1 Numerical explorations Versus the Parameter ? . . . . . . . . . . . . 65 3.3.2 Numerical explorations Versus the Parameter . . . . . . . . . . . . 67 3.3.3 Fractional order e¤ect on the localisation of Hopf bifurcation point . 69 4 Chaos Control of the fractional order systems 74 4.1 Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . 74 4.1.1 Ott-Grebogi-Yorke (OGY) Method . . . . . . . . .. 74 4.1.2 Feedback control . . . . . . . . . . . . . . . . . . 79 4.2 Feadbak Control of fractional Jerk System . . . . . . . 80 4.2.1 Description of the Model . . . . . . . . . . . . . . . . .0 4.2.2 Stability of the equilibrium points . . . . . . . . .81 4.2.3 Chaos control . . . . . . . . . . . . . . . 84 4.2.4 Numerical Results . . . . . . . . . . . . . 84 Conclusion 86 Bibliography 8 |
| En ligne : | http://thesis.univ-biskra.dz/id/eprint/6570 |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/161 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
