Normal Forms Computation And Bifurcation Analysis For Several Differential Equations

In this paper, we mainly study the dynamical property of three differential equation models, which include a predator-prey model with nonlinear harvesting, a delayed modified Leslie-Gower predator-prey model and an oscillator model with positive damping and two delays. By analyzing these three models from different point of view, we can obtain the unfolding normal forms and bifurcations at corresponding singularity for these models.In chapter one, we mainly introduce the background, current situation and some corresponding conceptions of mathematic ecology.In chapter two, we mainly study the multiple bifurcations of a ratio-dependent predator-prey model with nonlinear harvesting. In this model, we introduce nonlinear harvesting p(x)=qEx/m1E+m2x which is relative to prey. With the analyzation of the corresponding char-acteristic equation, the stability of boundary and interior equilibria can be obtained. By applying ordinary differential equation qualitative theory and differential manifold theo-rem, we can obtain the bifurcations at different equilibria.In chapter three, the bifurcations of a delayed modified Leslie-Gower predator-prey system are investigated. We first give the existence conditions that an equilibrium is Bogdanov-Takens (B-T) or triple-zero singularity. Then we present the unfolding normal forms and bifurcations at corresponding singularities by choosing appropriate bifurcation parameter and using center manifold reduction along with normal form theory. Last, the Hopf bifurcation of the system at another interior equilibrium is analyzed by taking delay as bifurcation parameter.In chapter four, the B-T bifurcation in an oscillator with positive damping and two delays is investigated. After discussing the distribution of the eigenvalues of the charac-teristic equations, we give the existence conditions which let the origin of the system be a B-T singularity. Furthermore, by means of center manifold theory, the second and third order normal forms of B-T bifurcation for the system are obtained.