On The Singular Points Of Polynomial Systems

Qualitative theory of ordinary diferential equations is an efective method tostudy nonlinear systems. In local qualitative analysis, singular points play a crucialrole, because in a neighbourhood of a normal point, the orbit structure is trivial, butusually it is complicated near a singular point. In this dissertation, which consistsof the following four parts, we study the qualitative theory of planar polynomialsystems at a singular point.The frst part, i.e. Chapter1, is an introduction to the dissertation. We presentthe development of singularity qualitative theory of planar polynomial systems andsummarize the main results of this dissertation.Chapter2as the second part provides preliminaries for the dissertation. Wefrst introduce normal form theory which is also the theoretical foundation of thisdissertation. This section includes not only classical linearization theorems, butalso some new results, such as the theorem on admissible nonlinearity. Then wegive the classifcation of elementary singular points of planar analytic systems andcorresponding normal forms, followed by generalizing the concept of a center tocomplex systems. In the fnal section, we present a practical method for integrabilityand linearizability, Darboux method.The third part including Chapter3and Chapter4is the most important one in this dissertation. In these two chapters, integrability and linearizability problemsof polynomial systems are studied specifcally, and the main results can be brieflysummed up as follows.In Chapter3, the linearizability problem is considered. It is well known thata linearizable center is equivalent to an isochronous center in R2. The concept ofisochronicity can be also extended to complex systems by the idea of linearizability.Then we mainly study homogeneous quartic polynomial systems with a1:′2resonant saddle, and obtain the necessary and sufcient conditions(divided into10cases) for linearizing them in a neighbourhood of the saddle point. Then thelinearizability problem of these systems is solved completely.Chapter4is devoted to fnding the highest possible saddle order of noninte-grable systems, an accompanied problem of integrability. The saddle order problemof systems,x|.=px-y~n,, y|.=-qy+x~n, with general resonances and degrees at the o-rigin, is mainly studied. By associating these systems with a class of Lotka-Volterrasystems and establishing a relationship between their saddle orders, we prove thatabove systems can have a saddle order no less than n~2-1under some conditions.That means, for systems with resonance and degree n satisfying some conditions,the highest possible saddle order is no less than n~2-1. At the end of this chapter,we give an example, a cubic homogeneous Lotka-Volterra system having a saddleorder8at a1:-2resonant saddle.The last part contains three appendixes, which are the data of linearizabil-ity constants in Chapter3, tables on the present progresses in integrability andlinearizability problems, and necessary and sufcient conditions of integrability orlinearizability for some systems.