Les systémes chaotiques à dérivées fractionnaires

Abstract

This thesis deals with fractional-order chaotic systems. The main highlight is on some basic differences between a fractional-order system and its integer order counterpart. Namely, stability conditions, existence of periodic solutions and min-imal total order for which chaos can occur etc... The finding of a new chaotic attractor from Hybrid optical bistable system is reported and dynamic of the new system is investigated in both integer and fractional-order cases. It is shown that asymptotic stability of equilibrium points of the fractional system can occur with positive real part of some corresponding eigenvalues which is not the case in integer-order systems. We have established criterion under which a fractional-order system undergoes Hopf bifurcation. The results are validated by mean of stability theory and numerical simulations. It is shown that chaos can be occurred in fractional-order system with total order less than three which is not the case in integer-order system due to the Poincar´e- Bendixon theorem. Finally, nonlinear feedback control scheme has been extended to control fractional financial system. The results are proved analytically by applying the stability condition for fractional system. Numerically the unstable fixed points have been successively stabilized for different values of fractional order; moreover some un-stable periodic orbits have been stabilized.