Global Asymptotic Stability For The Higher Order Rational Difference Equations

Rational, difference equations appear as discrete analogues and numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology,physiology,physics,engineering and economics,whose qualitative analysis has always been the hot spot subject to be studied in recent years. In this dissertation we mainly consider global asymptotical stability of three kinds of rational difference equation.Recently there has been a great interest in studying the qualitative properties of rational difference equations.Some prototypes for the development of the basic theory of the global behavior of nonlinear difference equations of order greater than one come from the results for rational difference equations.However,there have not been any effective general methods to deal with the global behavior of rational difference equations of order greater than one so far. From the known work,one can see that it is extremely difficult to understand thoroughly the behavior of solutions of rational difference equations although they simple forms.The method of Lyapunov is also a effective general method,Therefore, the study of the qualitative properties of rational difference equations of order greater than one is worth further consideration.First,in Chapter 2, the global asymptotical stability for a forth order rational difference equations was investigated by using "Semi-cycle Analysis Method",the method which can be used to solve the global asymptotic stability for a kind of rational difference equation,but it is difficult to solve the higher order rational difference equations directly. Some known results are generalized.Second,the global asymptotic stability for a higher order rational difference equation was investigated by using "Subsequence Analysis Method " in Chapter 3,and the "Semi-cycle Analysis Method" is a extension and supplement of "Semi-cycle Analysis Method". A set of sufficient conditions for the global stability of higher order rational difference equations are also obtained.Finally, with help of a auxiliary equation,a set of sufficient conditions for the global stability of higher order rational difference equations were obtained.We also give some equations in each chapter of dissertation,Whose global asymptotic stability can be proved by the same method.