Analyse du chaos dans un système d’équations différentielles fractionnaires

Abstract

In this thesis, we focus on the differential equations of fractional order systems exhibiting chaotic dynamics. Particular attention has been paid to a nonlinear system of fractional differential equations modeling the phenomenon of nuclear magnetic resonance : the Bloch system. A qualitative analysis of the dynamics of this system has been made, including some basic properties : bifurcations, periodic windows and routes to chaos. These properties were analyzed numerically by bifurcation diagram, phase portraits and Lyapunov exponents . The chaotic behavior of this system was confirmed by the existence of a positive Lyapunov exponent. On the other hand, the topological horseshoe was found, rigorously proving the chaotic nature of our system for certain parameter values, this method is considered as an excellent substitute for the Lyapunov spectrum method, less reliable numerically. 0-1 test provides a simple and efficient criterion for the distinction chaotic solutions of regular orbits, we have successfully applied this test in our work. Finally, the method of non-linear control was extended to realize the identical synchronization of two fractional Bloch systems. The results were proven analytically using stability conditions for fractional systems. A numerical simulation was performed to validate the results.