一类含有二次项的三阶有理差分方程的全局行为

Global behaviours of a third-order rational difference equations with quadratic terms

摘要:

本文给出方程\,$x_{n+1}=-\frac{x_n x_{n-1}}{ ax_n+bx_{n-2}}$\,的奇点集, 并讨论该方程解的全局行为. 我们证明了收敛解要么是一个4周期解, 要么收敛到一个4周期解或固定值; 不收敛解在一定条件下是无界的.

Abstract:

In this paper, we discuss the global behavior of a rational difference equation\,$x_{n+1}=-\frac{x_n x_{n-1}}{ ax_n+bx_{n-2}}$\,. It shows that for all initial values outside the forbidden set, its convergence solution is either a 4 periodic solution, or converges to a 4 periodic solution or a fixed value; the non-convergence solution is unbounded under some conditions.