This paper contains some sufficient condition for the point zero to be a global attractor for nonlinear recurrence of second order.

In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation $$x_{n+1}=\frac{a{x}_n{x}_{n-1}}{-b{x}_n+c{x}_{n-2}},n\in {\mathbb{N}}_0$$ where $a$, $b$, $c$ are positive real numbers and the initial conditions $x_{-2}$, $x_{-1}$, $x_0$ are real numbers. We show that every admissible solution of that equation converges to zero if either $a<c$ or $a>c$ with $(a-c)/b<1$. When $a>c$ with $(a-c)/b>1$, we prove that every admissible solution is unbounded. Finally, when $a=c$, we prove that every admissible solution converges to zero.

In this paper, we introduce an explicit formula and discuss the global behavior of solutions of the difference equation $$x_{n+1}=\frac{a{x}_{n-3}}{b+c{x}_{n-1}{x}_{n-3}},n=0,1,\cdots$$ where $a,b,c$ are positive real numbers and the initial conditions $x_{-3}$, $x_{-2}$, $x_{-1}$, $x_0$ are real numbers.