Bifurcations And Phase Portraits Of The Quadratic Hamiltonian System On Sphere

With the persistent improvement of the ability of human being of the recognizing ,reconstructing and utilizing the nature ,as well as the needs of practical apply, people have to face to deal with huge nonlinear problems. Hamiltonian system is an impor-tant field of nonlinear science study, whose creation and development have a profound practical background.As a kind of generalization of classical Hamiltonian system, generalized Hamilton system can be used to study many nonlinear dynamical models with any dimension including odd dimension, which come from divers science fields, such as mechanical engineering, optics, and molecular dynamics,et al.From the existing results of practical models , we note that generalized Hamilton system with sphere foliation structure and quadratic Hamiltonian function often appears as dynamical models of many real problems. It has been shown that such systems can be reduced the following five cases: Previous researches had devoted to the first four cases and obtained the complete re-sults on the corresponding Hamiltonian systems. Up to the present,however, to Hamil-ton function with four-parameters, the bifurcation properties and phase portraits of the corresponding systems are yet to be studied.Based on the facts mentioned above , by making use of the qualitative theories of differential equation and the bifurcation theory of dynamics system (especially Hamil-tonian system phase diagram analyze technique), this paper study the four-parameter Hamiltonian system (the fifth case) and analyze the bifurcation and stability of equilib-ria in detail, obtain the classification of phase portraits on sphere.