Stability and bifurcation analysis of maxwell- bloch equations

Abstract

In this thesis, stability and bifurcation analysis of Maxwell-Bloch equations were considered. By the aid of divergence test, it was proved that the system is dissipative. Steady state points of the equations were determined. The equations were linearized using Jacobian matrix about each equilibrium points. The local stability condition of each critical point was proved by using Routh- Huwertiz stability criteria. By the aid of Lyapunov theorem, equilibrium point one was proved to be globally asymptotically stable with some specific condition on pumping energy parameter. It is impossible to speak global stability property of the two remaining equilibrium points in the sense of Lyapunov due to the fact that one of the criteria to apply the theorem is not satisfied. Furthermore, the result of Hopf bifurcation revealed that the system doesn’t undergo Hopf bifurcation at equilibrium point one by any choice of pumping energy parameter and with some specific conditions the system undergoes Hopf bifurcation about the two remaining equilibrium points for a certain value of pumping energy parameter. Finally, in order to verify the applicability of the result two numerical examples were solved and MATLAB simulation was implemented to support the findings of the study