一类二阶有理差分方程的解的渐近性

Asymptotic Behavior of Solutions for a Class of a Two-Order Rational Difference Equation

用差分方程的动力学定理、Routh-Hurwitz和Schur-Cohn判别法、计算机Matlab数值计算这三种不同的方法研究了二阶有理差分方程x_(n+1)=ax_(n-1)+x_(n)x_(n-1)/bx_(n)+c(n-1)+d,n=0,1,2,…,的解{x_(n)}^(∞)_(n=-1)的渐近性,其中a,b,c,d∈R^(+),初始值x_(-1),x_(0)∈R^(+).并由a,b,c,d的取值的不同,得到不同的解的渐近性,并给出了平衡解是汇点、源点、鞍点、非双曲点的充要条件.

The asymptotic behavior of the solution of second-order rational difference equation x_(n+1)=ax_(n-1)+x_(n)x_(n-1)/bx_(n)+c(n-1)+d,n=0,1,2,…,(a,b,c,d∈R^(+),x_(-1),x_(0)∈R^(+))is studied by three different methods:the dynamic theorem of the difference equation,the Routh Hurwitz and Schur Cohn discriminant method and the numerical calculation of computer MATLAB.From the different values of a,b,c,d,the asymptotic properties of different solutions are obtained,and the necessary and sufficient conditions for the equilibrium solution to be a sink point,a source point,a saddle point and a non hyperbolic point are given.