Stability Of Periodic Solutions For Nonlinear Difference Equations And Applications

The difference equation is a discrete mathematical model of changing over time, in reality, the discrete phenomenon change everywhere. It has been around for hundreds of years of history, in which rational difference equation has attracted many researchers in the world. With the increasing developmen-t of difference equation, stability, asymptotic behavior, bifurcation and the optimal control problem has aroused many mathematicians’attention. The difference equation has a very wide range of applications, in addition to the field of mathematics, it also appeared in physics, biology, automatic control, population dynamics, economics, medicine and many other disciplines and branches. Therefore, based on the widely used in these fields, there has been an increasing interest in the study of the stability of the solutions for linear difference equation and no-linear difference equation in theory.This paper mainly studies the stability of periodic solution for nonlinear rational difference equations and application. Using the stable manifold theo-rem and related theory, the stabihty results of a class of two periodic solutions of two order rational difference equation are given. The asymptotic behav-ior、the stability problems of a class of three order differential equations are discussed. Finally, we give an application, to determine the optimal strategy of fishing for discrete optimal control.The paper is made up of four chapters. Main contents are as follows: In chapter one, we give a survey to the development and current state of the difference equation. We also introduce some preliminary material, includ-ing some basic theory and contents of difference equation.In chapter two, we study the local stability of period two positive solutions of second order rational difference equation where A, b>0,a≥0are real numbers, and the initial condition x-1and x0are arbitrary positive real numbers. The conclusion is that, the period two positive solutions is unstable saddle point. Stable manifold theorem is the main method. Our method is efficient for local stability of period two solutions of this second order rational difference equation and our conclusion has not been given before.In chapter three, we consider the third order rational difference equation where the initial conditions (x-2, x-1, x0)∈R3and the parameters a, b are ar-bitrary real numbers. We investigate the asymptotic behaviour and the stabil-ity properties of solutions of the above mentioned difference equation. Firstly, we derive the representation of the solutions for this equation. Secondly, we describe the asymptotic behaviour of the solution, this section is divided into three subsections depending on the values of a. Finally, we devote to analyze the stability properties of the periodic solutions of the third order rational difference equation.In chapter four, we give an application. Consideration was given to the discrete optimal control method for the optimal fishing strategy. Our method is new and efficient for discrete optimal control problem, which is different from the other optimal methods such as the traditional variational method, the Pontryagin principle of maximum and the dynamic programming. The basic construction of the model is the traditional logistic function relating to the growth of fry. The discrete optimal control method for optimal fishing strategy was used to construct the optimal rate of each fishing, the main focus of our work is on the rigorous mathematical analysis of the optimal control problem. The analysis allows one to obtain the optimal initial investment amount of the fry and the optimal size of the total catch. Furthermore, when the initial investment amount of the fry is below or above the optimal value, and the intrinsic growth rate of fish R is too small, we derive that fishing operations should not be started in the last few years to make the overall fishing amount optimal. At last, several typical examples are given to illustrated the obtained results.