Qualitative behavior of some rational difference equations (Q2794914)

The authors consider the rational difference equations NEWLINE\[NEWLINEx_{n+1} = {{x_{n-1}x_{n-4}}\over{x_{n-2}(\pm1\pm x_{n-1}x_{n-4})}} \;,\;n=0,1,\ldotsNEWLINE\]NEWLINE with arbitrary real numbers as initial conditions. The results strongly rely on the sufficient stability condition NEWLINE\[NEWLINE\sum_{i=1}^k|p_i|<1NEWLINE\]NEWLINE for the linear difference equation NEWLINE\[NEWLINEx_{n+k} + \sum_{i=1}^kp_ix_{n+k-i} = 0.NEWLINE\]NEWLINE For each of the four aforementioned rational difference equations they give the general expressions of the solutions, the equilibria and their stability/instability.