Pseudo Rotating Periodic Solutions For Delay Differential Systems

We give a proof of the existence and uniqueness of pseudo rotating periodic solutions for delay differential systems by exponential dichotomy and fixed point theorem.As is known to all,delay differential systems have been widely used in many fields,so it is of great importance to study the properties of their solutions.In this paper,we will exactly study pseudo rotating periodic solutions for finite or infinite delay differential systems by exponential dichotomy and fixed point theory.This paper is divided into six chapters.The first chapter introduces the background of delay differential systems,the definition of exponential dichotomy,and the relation between bounded solutions and exponential dichotomy;then the definition of rotational periodic function and pseudo rotating periodic function are given;finally,the main results of this paper are exactly described――the existence and uniqueness of the pseudo rotating periodic solutions for linear,semi-linear and nonlinear differential systems with finite delay and nonlinear differential dynamical systems with infinite delay.In chapter two,using exponential dichotomy,the principle of compression image and the Schauder fixed point theorem,we give a proof of the existence and uniqueness of rotating periodic solutions for linear,semi-linear and nonlinear differential systems with finite delay.On the basis of the second chapter,according to the definition of pseudo rotating periodic function,the third chapter studies the existence and uniqueness of pseudo rotating periodic solutions for linear,semi-linear and nonlinear differential systems with finite delay by similar method as chapter two.As an extension of finite delay differential systems,the existence and uniqueness of pseudo rotating periodic solutions of infinite delay differential systems are studied by two methods in chapter four.Its central idea is to use finite delay differential systems to approximate the infinite delay differential systems and prove the existence and uniqueness of the pseudo rotating periodic solution of infinite delay differential systems by the method similar to chapter three.Chapter five verifies the theorems by two examples.In the sixth chapter,we make a summary of this paper.