The Study Of Bifurcation And Chaos Of The Maxwell-Bloch Equations

The thesis deals with the dynamic behavior for a class of3-dimensional Maxwell-Bloch systems by using the method of dynamical system,including the type and stabil-ity analysis of the equilibria,Hopf bifurcation analysis,existence of Homoclinic Orbit and chaotic dynamical analysis and so on.Maxwell-Bloch system is very important class of nonlinear dynamical systems. There are many value of research results for this system in recent years.This class of system has been widely applied in many physi-cal areas,such as mechanical engineering,optics,and molecular dynamics. we give the system as follows In order to facilitate the analysis,this paper mainly studies the simplified three-dimensional model which is obtained by the Maxwell-Bloch system.We give the simplified system as follows First,we get that the simplified system have three equilibrium points when parameter η3>η1η2is hold.This three equilibrium points are A(0.0.773), and equilibrium points B、C is symmetrical.However,there is only one equilibrium point A when pa-rameter η3≤η1η2is satisfied.For the equilibrium point A,through theoretical proof we can see that it is stable when parameter η3≤η1η2hold,but it is unstable when param-eter773>771772is satisfied. For any η1>0,η2>0,η3>η1η2,equilibrium points B、C are asymptotically stable when and only when parameter and,but are unstable when Accord-ing to Hopf bifurcation theory,we give the theoretical analysis showing that the Hopf bifurcation occurs at the equilibrium B、C when the parameters given under the same conditions and we can get the subcritical Hopf bifurcation.In order to illustrate the re-liability of theoretical analysis,we set parameters η1=4,η2=1,η3as a free parameter in this paper. We can see that equilibrium points B、C occur simultaneously subcrit-ical Hopf bifurcation and we draw the corresponding bifurcation diagram used by the software Matcont.In this paper,we also convert the Maxwell-Bloch system to a system which is a generalized Hamiltonian systems with three dimension by a appropriate transforma-tion. This system can also be rewritten as a generalized Hamiltonian systems which is case of no disturbance and with the addition of the system after disturbance.For the generalized Hamiltonian system without disturbance,the orbits under various paramet-ric regions can be expressed as exact solutions by Jacobi elliptic function or hyperbolic function.Furthermore,we study the existence of homoclinic orbit and chaotic nature of the system in the addition of disturbance.By calculating the corresponding Melnikov function,we find that the homoclinic orbits still exist when the system is in the addi-tion of disturbance.Last but not least,by calculating the largest Lyapunov exponents,we notice that the system will be chaotic in a given parameters.