The rule of trajectory structure and global asymptotic stability for a nonlinear difference equation (Q5920462)

The author considers the fourth order rational difference equation \[ x_{n+1} = {{x_n(x_{n-2})^b + (x_{n-3})^b + a}\over{(x_{n-2})^b+x_n(x_{n-3})^b+a}}\;,\;n=0,1,2,\ldots\;,\;a\geq 0,\;b\geq 0 \] with strictly positive initial conditions \(x_i>0\), \(i=-3,-2,-1,0\). Four types of solutions are considered: positive and negative semicycles, eventually trivial and oscillatory. It is shown that the successive lengths of positive and negative semicycles for the nontrivial solutions occur periodically with prime period 15. Moreover the unique positive equilibrium is globally asymptotically stable.