Generalized Normal Sectors Method For Degenerate Equilibria And Related Bifurcations

Qualitative theory, which directly judges the qualitative properties of solutions by ordinary differential equations(ODEs) themselves, is very important in the study of ODEs. The ideas of qualitative theory influence other mathematical branches gradually. In particular, plentiful results are obtained for planar systems. Analysis of equilibria for planar systems is an important task in the qualitative theory. The discussion of degenerate equilibria is complicated.There are several ways to investigate the qualitative properties of degenerate equilibria. One often applies the method of Briot-Bouquet's transformations which decompose a complicated equilibrium into several simple ones, the method of Z-sectors and the method of normal sectors. However, these methods have weakness for some higher degenerate equilibria. For example, it is difficult to use Briot-Bouquet's transformations if the lowest degree of nonzero terms in a polynomial differential system is very high or even an unspecified integer. A Z-sector possibly contains more than one exceptional directions, so we can not determine which exceptional direction an orbit in the sector will be tangent to. On the other hand, neither a radial line nor an exceptional direction can be the edges of a normal sector, and that greatly confines the application of normal sectors. In chapter 2, we develop the idea of normal sectors to a new method of generalized normal sectors(GNS for short), which can be applied to analyze the structure of orbits near degenerate equilibria when above methods can not be useful. Furthermore, a GNS is a quasi-sectorial region, which does not restrict edges of the sectors to be zero branches of some function, wherefore it is easy to ascertain edges. A GNS includes at most one exceptional direction, thus we can easily judge the tangent direction of a orbit if it exist in this GNS. In addition, compared to a normal sector, a GNS has fewer restrictive conditions, as it does not restrict edges of the sectors to be radial lines but even allow orbits to be edges. In fact, a normal sector is a special case of a GNS. Therefore, more extensive class of degenerate equilibria can be discussed for qualitative properties by using the method of GNS.In Chapter 3 we use the method of GNS to investigate the high degenerate equilibria of some polynomial differential systems which have practical backgrounds, including a multi-molecular reaction system at infinity, a generalized Brusselator system atinfinity and a ratio dependent predator-prey system. The equilibria in these systems are high degenerate, so it is difficult to or can not use the methods of Briot-Bouquet's transformations, normal sectors or Z-sectors.Another important aspect about degenerate equilibria of vector fields is bifurcation. As parameters of a differential system are varied, changes may occur in the qualitative structure of the solutions for certain parameter values. These changes are called bifurcations. If bifurcations or even a versal unfolding are given, all physical phenomena of this system are clear. The difficult cases are bifurcations of codimension > 2 and non-local bifurcations.Some local bifurcations such as Hopf bifurcation and saddle-node bifurcation are discussed for a general multi-molecular reaction system in [Comput. Math. Appl., 43(2002):1407-1423]. In Chapter 4 we investigate the remainder case, i.e., a Bogdanov-Takens bifurcation of codimension 2 at a cusp. The corresponding universal unfolding is given and the problem of its local bifurcations is solved completely.Recently many attentions are paid to the qualitative properties of a general ratio-dependent predator-prey system. The reference [J. Math. Biol. 42(2001): 489-506] conjectures the existence of a heteroclinic loop and claims it as an open problem. This problem was dealt with in many papers. [J. Math. Biol. 43(2001): 221-246] proves the existence of the heteroclinic loop by numerical approaches together with methods of analysis. However, no bifurcation values of parameters were given for existence of the heteroclinic loop. In chapter 5, we reduce this system to a perturbed Hamiltonian system so that Melnikov's method is available to study the heteroclinic bifurcation. We give parametric conditions of the existence of the heteroclinic loop analytically and answer the open problem further.