Global behavior of a three-dimensional linear fractional system of difference equations (Q2568206)

Consider the system of difference equations \[ \begin{aligned} x_{n+1} &= (a+X_n)/(b+y_n),\\ y_{n+1}&=(c+y_n)/(d+z_n),\quad n=0,1,\dots,\\ z_{n+1}&=(e+z_n)/(f+x_n).\end{aligned} \] If all solutions to the characteristic equation \[ \lambda^3-r \lambda^2-s \lambda-t=0 \] of the Jacobian of the map \[ T(x,y,z)=\bigl[(a+x) / (b+y),(c+y)/(d+z),(e+z)/(f+x)\bigr] \] lie inside of the open disc \(|\lambda |<1\), then the equilibrium of the system is locally asymptotically stable, and if at least one of them lies outside a closed unit disc then the equilibrium is unstable. The global attractivity is also considered.