The Geometric Theory Of Quasi-homogeneous Vector Fields And Related Problems Research

It is well known that the planar dynamical systems have abundant results on both theory and the methods, while there is no general theory for the nonlinear systems in three dimensional Euclidean space R3. If we can construct a bridge between planar dynamical system and dynamical system in R3, then we can provide many methods and tools for research dynamical systems in R3. The simpler vector fields in R3 is homogeneous vector fields, which already has many results. In this paper, we analysis quasi-homogeneous vector fields with the methods of homogeneous vector fields, and then we give the application of quasi-homogeneous vector fields in biomathematics. The paper consist of two chapters, the main contents of each chapter are as following:Chapter 1 discussed the geometric properties of quasi-homogeneous vector fields in R3.By the tangent vector fields introduced in sphere S2 we got the necessary and sufficient conditions for the existence of the closed orbits in the vector fields in R3.and we can know the closed orbits located in a closed invariant cones. This chapter also gave the application of quasi-homogeneous vector fields in biomathematics, and obtains the results that the necessary and sufficient conditions for the existence and uniqueness of the globally attractive periodic solution in the space of first quadrant with a perturbation model.Chapter 2 discussed a class of prey with constant harvest and with Holling-IV type functional response predator-prey models by using the qualitative theory in differential equations. We researched the behavior of the equilibrium points, and got the existence of the saddle-node bifurcation,Hopf bifurcation and nilpotent saddle bifurcation and the normal form for unfolding the nilpotent saddle. Then we obtained the first two Lyapunov coefficients of the model by the formula, and the second is always positive. At last we gave the biological interpretation of the model at the above conditions.