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

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.