Analytic Invariant Curves And Analytic Solutions Of Iteratives Functions Equation

The purpose of dynamical system theory is to study rules of change in state which depends on time.Usually there are two basic of dynamical systems:continuous dynamical systems by differential equations and discrete dynamical systems described by iteration of mappings.Many mathematical models in physics,mechanics,biology and astronomy are given in such forma.Many problems of dynamical systems can be reduced to an iterative functional equation.Through years of development,iterative functional equations have become a branch of modern mathematics that are closely related to differential equations,difference equations,integral equations and dynamical systems,playing an important role in the study of experimental science and engineering.Since mathematicians like Babbage,Abel etc.iterative functional equations have formed a theory system with the development of iterative theory.In the introduction part of the thesis,the characteristics and applications of functional iteration,the concepts of iterations and dynamical systems,theconcepts of discrete dynamical systems and continuous dynamical systems,the basic forms of iterative euations and the problems of iterative roots and invariant curves and Davie lemma are introduced.A brief introduction is also given about the achievements made in the field of iterative functional equations in recent years.As an important model abstracted form the real world,iterative functional equations are of wide operation significance and application background,being always concerned by mathematicians.In experiment,the analysis of the regulation of system sport is always carried out by means of the records made form initialization to current state.Iterative functional equations are the outcome of function compound and iteration,and,just like differential equations,are a special type of function equatons.Accurately speaking,iterative functional equations are the identical equation formed by unknown functions and compound operation.We reduce the existence of analytic invariant curves to the existence of an iterative functional equation by means of majorant series.It play an important role in the theory of periodic stability of discrete dynamical systems.In chapter 2,three kinds of planar mappings are discussed.We reduce the existence of analytic invariant curves of iterative functional equation by means of majorant series.Previous works requireα,the eigenvalue of the linearization of the unknown function at its fixed point,is not on the unit circle or lies on the circle with the Diophantine condition.We break through the restriction of Diophantine condition and obtain results of analytic solutions in the case of unit rootαand the case thatαlies on the circle but doesn't satisfied with the Diophantine condition,using the weaker condition than Diophantine condition—Brjuno condition.The generation is a kind of medium widespread phenomenon of nature and the mankind live.The generation is suffused with the letter differential calculus square distance and often the differential calculus square distance contain very great dissimilarity.The hour of this kind of square distance not only depends on in time but also depends on in the appearance even the appearance lead a number.Get into 80's, people discovered the various application of this kind of square distance more and more.For example,the importance that is in all putting forwardt he square distance of this kind of type in the physics,control theory,the a series of problems such as theory and biology etc.displaying them at the application and theoretically,also stir up to have people thus to their mightiness of interest in the research.In order to doing not know function the emergence of the generation.Often the existence axioms of classic in the differential calculus square distance can't use.Textual chapter 3 makes use of transformation to change into a generation square distance not to contain to don't know function generation not the distance with square differential calculus of the line generation,making use of again an excellent series meteod to get existence of analyzing the solution.