Researches On Classification For Three-dimensional Quadratic Hamiltonian Systems And Related Problems

The thesis deals with the structure-preserving transformations of three-dimensional real Lie Algebras, and the simplification and classification of quadratic Hamiltonians. On this base, for the quadratic Hamiltonian systems on the dual space of a class of three-dimensional real Lie Algebra, the number and stability of its equilibria, the foliated structures and the bifurcation of parameters arc analyzed, which gained the complete global phase portraits. In addition, a class of three-dimensional generalized Hamilto-nian system with a nonlinear Poisson structure and quadratic Hamiltonian function and pinched sphere leaves has been introduced. For which, we make detailed and thorough analysis the number of equilibria, the existence and stability conditions of equilibrium states, the foliated structures, the bifurcation of parameters and the global phase por-traits. Meanwhile some interested results arc obtained.This thesis has three chapters.In chapter1. we briefly introduce the research background and motivation of this research. Some definitions and propositions to be used are also introduced in this chapter.In chapter2, based on the famous Bianchi Classification, we study the structure-preserving transformations of three-dimensional real Lie Algebras. By moans of the theory of generalized Hamiltonian systems and mathematical software Maple, all linear trans-formations which preserve the Lie-Poisson structure on three-dimensional space are ob-tained. By making appropriate transformation and the normal form theory, we discuss the simplification and elassification of quadratic Hanriltonians.In chapter3, we consider the bifurcation analysis and phase portraits of a class of three-dimensional generalized Hamiltonian system quadratic Hamiltonian with a nonlin-ear Poisson structure and quadratic Hamiltonian function. With the help of the qualita-tive theory of ordinary differential equation, the bifurcation theory of dynamical systems and the theory of generalized Hamiltonian systems, we obtain the number of equilibria, the existence and stability conditions of equilibrium states, the foliated structures, the bifurcation of parameters, the analysis of global phase portraits of this system.