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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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