Small Divisors Problem In Dynamical Systems And Linearization Of Germs And Vector Fields

Form the end of the 19th century to the beginning of the 20th century, Poincare and others proposed the concept of dynamic system from the study of classical me-chanics and qualitative theory of differential equations. Modern study of dynamical systems by Peixoto and others began in the early 60s of the 20th century. The basic theory of this discipline has made significant progress under Smale and many other scholars's advocacy and promotion. Since the 70s of the 20th century, dy-namical systems have been applied to the wider areas and also show a wide ap-plication prospect, such as economic mathematics, weather forecasting, numerical computation, statistical mechanics, vibration theory, chemical reactions, physiologi-cal processes, economic and demographic issues and so on. This subject has received wide attentions is not only because of its rich and in-depth theory, but slao because its broad and effective application has become a hot research spot of the nonlinear science.This article dicusses the small divisor problem in dynamic systems, the analytic solutions and smooth solutions of the Shabat equation and the linearization problem of analytic germs and vector fields.In chapterâ… , we introduce the development and basic concepts of small divisors problem and linearization problems, which provide the necessary theoretical basis for the proof of the second, third, fourth, five chapters.In chapter 2 and chapter 3, we discuss the analytic solutions and smooth solutions of the Shabat equation and the Shabat-like equation. We prove that the existence of analytic solutions and smooth solutions of the Shabat equation under the Brjuno condition, which generalized the conclusion of Y. Liu [43]. We improved Davies lemma and find a new arithmetical condition which is weaker than the Brjuno condition, and also show the existence of Gevrey-like classes solutions of the Shabat equation and the Shabat-like equation under the new arithmetical condition.In chapter 4, we study the linearization problem analytic germs with the quasi parabolic fixed point. Due to the presence of resonances, this problem becomes more difficult. we generalized the result of F. Rong [56]. In chapter 5, we study the linearization problem of the quasi parabolic vector fields, which extends the conclusion from analytic germs to analytic vector fields.