Qualitative Analysis Of Polynomial Poincaré Equation And Existence Of Periodic Solutions Of A Class Of Ecology Equations

This thesis investigates the focus-center problem of the quartic polynomial Poincare equation, the problem of Hopf's bifurcation of a singular point of focus-center type for the cubic and quartic polynomial Poincare' equation, and the problem of existence of periodic solutions for a class of two-dimensional biological model with periodic coefficients. Whole paper is composed of four chapters.The chapter one is the introduction, in which we introduce the relevant investigation and development of the qualitative theory of ordinary defferential equation, and briefly represent the main works of this thesis.In the chapter two, by use of a kind of linear recurrence formula for computing the focal value, we work out the distinguishing between focus and center for two classes of quartic polynomial Poincare' equations, and determining their highest order of focus. By computing focal values of quartic polynomial Poincare' equation with full coefficients, we give a restrictive condition among the coefficients, under which such an equation has focus of order 7 at the origin, further conjecture that the highest order of the focus for the quartic polynomial Poincare equation with full coefficients is 8. The strict proof of the conjecture perheps remains to be studied long.In the chapter three, we study the problem of Hopf's bifurcation of a singular point of focus-center type for the cubic and quartic polynomial Poincare equation by small perturbing its coefficients, which is compared with Pugh's problem. For the cubic polynomial Poincare equation, by means of a known result about Pugh's problem, we discuss the problem of the uniqueness and only-two of limit cycles. For the quartic polynomial Poincare equation, on the basis of the investigation of the chapter two, we give an example with six small limit cycles, and propose to compare the example with Nirenberg's example, of which Smale spoke.In the chapter four, we discuss three two-dimensional predator-prey medols with exploited terms and periodic coefficients. By using the Mawhin's continuation theorem of coincidence degree theory, we give conditions for such three medols to have at least one, two and four periodic solutions respectively. It seems to be seldom seen in the same kind of investigations, which ones derive the existence of several periodic solutions by the continuation theory of coincidence degree.