The Normal Forms Of Degenerate Analytic Systems And Applications

Normal forms are powerfully analytical tools for studying the qualitative properties of nonlinear dynamical systems.The basic idea of the normal form theory is how to look for the right variable transformation,and in keeping the two systems before and after the transformation of local qualitative properties unchanged at the same time,then make the system after transformation in form as simple as possible,in order to maximize the simplified nonlinear systems in the field of the singularity of the local dynamics analysis.The computation of a normal form for the given nonlinear dynamical systems is one of main contents in the theory of normal forms of nonlinear dynamical systems.But the calculation method of the existing normal form is mainly to calculate the coefficient matrix of the linear part of non-zero power system,and hard to get the near identity change of coordinates of the form at the same time.But,in the practical application,there are a lot of power systems which the coefficient matrices of the linear part is zero(degradation of the power system),need us to study normal form the degradation of the power system,then provide the basis for the qualitative study.So the calculation method of the degradation systems is of great practical significance.This dissertation promotes the non-degenerate Carleman linearization method to the degenerate system,and use the promotion to calculate a class of the second degenerate planar system’s normal form and the third degenerate planar system’s normal form,and at the same time the near identity change of coordinates of the form is given.The qualitative properties of the system for further analysis of this kind of degradation is established.Because the system is degenerate,it’s different from the research with non-zero coefficient matrix A of the power system,which only need to consider the influence of the near identity change of coordinates for the same item number,computing the normal forms of degenerate systems need to consider the influence of the transformation of higher order terms.The dissertation consists of four chapters:The first chapter is the introduction which mainly introduced the paper research background and significance,analyzed the research status at home and abroad,introduced the main content of the dissertation research,and recommend the structure and arrangement of this paper.The second chapter mainly presents the method of Carleman linearization,and its application in the calculating of a class of the second sub-degree degenerate planar system’s normal form,then we obtain the simplification of such a system and the near identity change of coordinates at the same time.The third chapter mainly presents the method of Carleman linearization,and its application in the calculating of a class of the third sub-degree degenerate planar system’s normal form,then we obtain simplification of such a system and the near identity change of coordinates at the same time.The fourth chapter makes a summary and outlook to full text,gives the results and some issues worthy of further research.