Dynamic Analysis Of Three-dimensional Maxwell-Bloch Systemy

As an extremely important effect in the semiclassical theory of laser,the interaction between the laser radiation field and atoms plays an important role not only in the functional circuits of lasers,but also in the interpretation of the physical phenomena of quantum optics.Their interaction,which is usually described by differential equations,is often understood and mastered through the in-depth analysis of the differential equations describing the interaction regularity and thus promotes the development of related theories in physics,mathematics and other fields.In this dissertation,the dynamic behavior of three-dimensional Maxwell-Bloch system,which is established based on the interaction between laser and atoms,are analyzed.The specific work are as follows:In Chapter 1,the research background,significance and status quo of this paper are expounded.The research status quo of Maxwell-Bloch system are sketched.The research situation of boundedness of system is briefly summarized.The advanced studies of Poincaré compactification technology and Jacobi stability are introduced.In Chapter 2,the mathematical model of three-dimensional Maxwell-Bloch ordinary differential equation is simply derived;the ultimate bound set of three-dimensional system is obtained by using the optimization method and the Lagrange multiplier method;the characteristics of the infinite singularities of three-dimensional system are analyzed,based on Poincaré compactification technique.The topological structure of the system at infinity is portrayed,and the obtaining results show that the dynamic behavior of singularities at the infinity of the x-axis?y-axis and z-axis are very complex in combination with numerical simulation.In Chapter 3,the closed orbit properties,the existence of homoclinic orbits and singularly degenerate heteroclinic cycles for three-dimensional Maxwell-Bloch system are considered.By using the surface classification theory of quadric surface theory in three-dimensional space,it is proven that the closed orbit of the system can not fall on the same plane.Based on the generalized Melnikov method,two nontransverse homoclinnic orbits are strictly proved to exist under the conditions of some parameters(cd is sufficiently large).Contrarily,it is proved analytically that there exists an infinite set of singularly degenerate heteroclinic cycles in the system when the parameter c is sufficiently small(c = 0).The chaotic mechanism of the system is further explored by numerical simulation,and the obtaining results show that the system moves towards chaos by mean of the rupture of singularly degenerate heteroclinic cycles with the change of parameters.In Chapter 4,the Jacobi stability of three-dimensional Maxwell-Bloch system is considered based on KCC theory.By calculating five geometric invariants of a secondorder system,Jacobi stability parameter conditions on the trajectories of the system are obtained.With the help of introducing instability exponent and curvature,the chaotic mechanism of the system is analyzed tentatively in combination with numerical simulation.