The Normal Form And Its Application Of Analytic System

The basic idea of the normal form theory is how to look for a simpler formal differentialsystem than the given nonlinear differential system, and, in the meantime, to keep its essentialfeatures invariant, that is, the simpler differential system that we got is equivalent to the originaldifferential system in some meaning. It is a problem that we must face to check whether twodifferential systems are equivalent. As we know, this equivalence is defined as the two systemshave the same topological structures. However, it is obvious that the topological structure isdifferent from the qualitative structure, and the latter one can show finer dynamical behaviors ofa nonlinear differential system than previous one. For example, the saddle point of a planarnon-degenerate linear system has the same topological structure with the focal point, but theirdynamical behaviors shows clearly different. Till now, we have not found that the definition ofthe qualitative structure of a planar differential system or a vector field, even for an analytic case.In this dissertation, we give the definition of the qualitative structure of a singular point of aplanar analytic system. In addition, as an example, we classify the singularities of planarnon-degenerate analytic systems according to the definition of the qualitative structure. Theresults show that the definition we give is reasonable, and it is better to classify singularities ofnon-degenerate planar analytic system to show the dynamical behaviors according to thequalitative structure than the topological structure.The normal form of a nonlinear differential system is generally not unique, so it issignificant to study the relationship of the different normal forms. In this dissertation, we givethe corresponding relationship of the monomial coefficients of two different normal forms of anilpotent system.The nilpotent system is a class of nonlinear differential systems with extensive applicationvalues. For example, when studying the existence of traveling wave solutions of a partialdifferential equation, by making a traveling wave change, the partial differential equation isusually transformed to a nilpotent system of ordinary differential equation. In this dissertation,we study the monodromy of singularity of a nilpotent system by normal form theory andquasi-homogeneous polar coordinate Blow up.Finally, we make a brief summary and present some problems for future works.