Stability And Periodic Character Of A Rational Second Order Difference Equation

The difference equation describes a system’s regularities which are changing over time. Tt is a discrete mathematical model. In recent years, more and more math-ematicians all over the world start to research the rational difference equation. Be-cause of their simple forms, people have got many nice results which conclude the asymptotic behavior, the global behavior, the boundedness, the oscillation, the peri-odicity and so on. In the theory of the difference equation, the stability theory is more important. So it becomes a hot research issue among people.The paper mainly studies the stability of equilibrium point and the periodic so-lution for a rational difference equation. We get the conditions for the existence of periodic solutions of the positive period two. Then we obtain the explicit expression of the positive period two. At last, we discuss the stability of equilibrium point and the global behavior. The paper is made up of four chapters:In chapter one, we introduce the research background and status of the difference equation.In chapter two, firstly, we introduce some basic theories and contents;secondly, we mainly discuss the following rational difference equationWhere the parameters a, b are positive real numbers and the initial conditions z-1, x0 are positive real numbers.This chapter mainly discuss the stability of equilibrium point and the global be-havior, getting the following conclutions(1) Stability:When a< b< 3a or b< a< 3b, the equilibrium point x of equa-tion(0.2) is locally asymptotically stable.(2) Attractivity:The equilibrium point-x of equation(0.2) is a global attractor.Finally we combine the imaging to proceeding the numerical analysis, and verify the correctness of the conclusions.In chapter three, we mainly discuss the existence of periodic solutions of the positive period two, We get the explicit expression of the positive period two, that is Periodicity:When b> 3a> 0, the equation(0.2) exists a periodic solutions of posi-tive period two, that is: Finally we combine the imaging to proceeding the numerical analysis, and verify thecorrectness of the conclusions. In chapter four, we summarize the paper comprehensively and raise some ex-pects.