On some systems of difference equations (Q654636)

The authors consider the following three systems of nonlinear difference equations \[ u_{n+1}=\frac{v_n}{1+v_n},\quad v_{n+1}=\frac {u_n}{1+u_n}, \] \[ u_{n+1}=\frac{v_n}{1+u_n},\quad v_{n+1}=\frac {u_n}{1+v_n}, \] and \[ u_{n+1}=\frac{u_n}{1+v_n},\quad v_{n+1}=\frac {v_n}{1+u_n}, \] where \(n\in {\mathbb N}_0\) and the initial values \(u_0\) and \(v_0\) are given complex numbers. By help of the auxiliary Riccati equation \[ x_{n+1}=\frac{x_n}{a+bx_n}, \quad a,\,b\in \mathbb C,\,\,n\in {\mathbb N}_0, \] the general solution of each system is explicitly given. As a consequence, the asymptotic behaviour of the solutions is analysed in detail as \(n\rightarrow \infty\).