Dynamics of a three-dimensional systems of rational difference equations (Q2800514)

The authors study the existence of periodic solutions and the stability of the following systems of rational third-order difference equations NEWLINE\[NEWLINE x_{n+1} = \frac{y_n x_{n-2}}{x_{n-2} \pm z_{n-1}}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE y_{n+1} = \frac{z_n y_{n-2}}{y_{n-2} \pm x_{n-1}}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE z_{n+1} = \frac{x_n z_{n-2}}{z_{n-2} \pm y_{n-1}}, NEWLINE\]NEWLINE when nonzero real initial conditions \(x_{-2}, x_{-1}, x_0, y_{-2}, y_{-1}, y_0, z_{-2}, z_{-1}\), and \(z_0\) are specified. When the denominators are all of the form \(a_{n-2} + b_{n-1}\), they give the explicit solution of the system in terms of the Fibonacci sequence. They then show that the positive solutions of the system are bounded and converge to zero.NEWLINENEWLINEThey show that a similar result holds in the case where the denominators of two of the equations are of the form \(a_{n-2} + b_{n-1}\) and the third is of the form \(a_{n-2} - b_{n-1}\). In particular, they show that every solution is bounded and converges to zero.NEWLINENEWLINEWhen the denominator of two of the equations is of the form \(a_{n-2} - b_{n-1}\) and the third is of the form \(a_{n-2} + b_{n-1}\), they show that all solutions are periodic with period \(12\). As above, they give the explicit form of the solution.NEWLINENEWLINEThey also show that all solutions of the system NEWLINE\[NEWLINE x_{n+1} = \frac{y_n x_{n-2}}{x_{n-2} - z_{n-1}}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE y_{n+1} = \frac{z_n y_{n-2}}{y_{n-2} - x_{n-1}}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE z_{n+1} = \frac{x_n z_{n-2}}{z_{n-2} - y_{n-1}}, NEWLINE\]NEWLINE are periodic with period \(6\).