一类二阶有理差分方程的解的渐近性

Asymptotic Behavior of Solutions for a Class of Second-order Rational Difference Equation

本文使用Routh-Hurwitz判别法和Schur-Cohn判别法、有理差分方程的动力学理论等研究了二阶有理差分方程X_(n+1)=ax_(n)+x_(n-1)^(2)/(bx_(n)+cx_(n-1)+d),n=0,1,2,……的解{x_(n)}_(n=-1)^(∞)的渐近性,其中a,b,c,d∈R且b,c,d不同时为0,初始值x_(-1),x_(0)∈R,并由a,b,c,d的不同取值得到不同解的渐近性,同时给出平衡解是汇点、排斥点、鞍点、非双曲点等的充分条件。

We study the asymptotic behavior of the solution of the second-order rational difference equation X_(n+1)=ax_(n)+x_(n-1)^(2)/(bx_(n-1))(n=0,1,2,……,α,b,c,d∈R,b,c,d are not all zero,x_(-1),x_(0)∈R)by the different methods:the Routh Hurwitz discriminant method and Schur Cohn discriminant method and the dynamic theorem of the difference equation.From the different values of α,b,c,d∈R,the asymptotic properties of different solutions are obtained,and the sufficient conditions for the equilibrium solution to be a sink point,a repulsive point,a saddle point and a non hyperbolic point are given.