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Global behaviour of a second-order nonlinear difference equation. (English) Zbl 1232.39014

The article deals with the Cauchy problem \[ x_{n+1} = \frac{x_{n-1}}{a + bx_nx_{n-1}}, \qquad (x_{-1},x_0) \in {\mathbb R}^2, \;n = 0,1,2,\dots, \] where \(a, b\) are reals. It is proved that the general solution to this equation (beside some exceptional cases) is presented in the form \[ x_{2k+2} = x_0 \, \prod_{i=0}^k h(2i + 1), \quad x_{2k+1} = x_{-1} \, \prod_{i=0}^k h(2i), \] where \[ h(n) =\begin{cases} \frac{a^n(1 - a) + \alpha(-a^n)}{a^{n+1}(1 - a) + \alpha(1 - a^{n+1}}, & \text{if} \quad a \neq 1, \\ \frac{1 + \alpha n}{1 + \alpha(n + 1)}, & \text{if} \quad a = 1.\end{cases} \] (\(\alpha = bx_{-1}x_0\)). Further, a complete analysis of the asymptotic behavior of these solutions in different cases (\(a = -1\); \(|a| \geq 1, \;a \neq -1\); \(|a| < 1\)) is given, the behavior of these solutions near bifurcation points (\(a = -1\) and \(a = 1\)) s described and the stability properties of nonzero periodic solutions is investigated.

MSC:

39A20 Multiplicative and other generalized difference equations
39A30 Stability theory for difference equations
39A28 Bifurcation theory for difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
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