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On the dynamics of solutions of non-linear recursive system. (English) Zbl 1215.39014
The authors consider the difference system
$x_{n+1}={{A}\over{y_n}},\quad y_{n+1}={{Bx_{n-1}}\over{x_ny_{n-1}}},\quad A\neq 0 ,\;B\neq 0$
with non-zero initial conditions $$x_{-1}$$, $$x_0$$, $$y_{-1}$$, $$y_0$$. It is shown that all solutions are periodic with period six. Using mathematical induction for $$n$$, they obtain expressions of the solutions that correspond to the above initial conditions. The equilibria of the system are $$(A/\sqrt{B},\sqrt{B})$$ and $$(-A/\sqrt{B},-\sqrt{B})$$ and the characteristic equation of their Jacobian matrix is in both cases $$\lambda^2+\lambda=0$$, hence stability by the first approximation is not exponential.
##### MSC:
 39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type 39A12 Discrete version of topics in analysis 39A23 Periodic solutions of difference equations 39A30 Stability theory for difference equations
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