Theory Of Analytic Invariant Curves For A Planar Map

Nonlinear science is one of the most important topics in today's sciences. The theory of iterative dynamical systems plays an important role in nonlinear science.The purpose of dynamical system theory is to study rules of change in state which depends on time. Usually there are two basic forms of dynamical systems: continuous dynamical systems described by differential equations and discrete dynamical systems described by iteration of mappings. The study of iterative dynamical systems involves self-mappings on intervals, iterative roots of functions, iterative functional equations, iterative functional differential equations and embedding flows.Many mathematical models in physics, mechanics, biology and astronomy are given in such forms. Many problems of dynamical systems can be reduced to an iterative functional equation. Iterative equations are those equations which involve iteration as a basic operation, affect the development of natural science and engineering deeply. Since mathematicians like Babbage, Abel etc., iterative functional equations have formed a theory system with the development of iterative theory.In chapter one, the characteristics and applications of functional iteration, the concepts of iterations and dynamical systems, the concepts of discrete dynamical systems and continuous dynamical systems, the basic forms of iterative equations 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.Invariant curves of the area preserving maps play an important role in the theory of periodic Stability of discrete dynamical systems. In chapter two, we discuss the existence of analytic invariant curves for two kinds of planar mappings. We reduce the existence of invariant curves to the existence of an iterative functional equation. Then we use the Schroder transformation to change the iterative functional equation to another without iterates of the unknown function. Further, we obtain the existence of analytic solutions of such an 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 that given functions have a regular singular Point.