## 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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### References:

 [1] Agarwal R.P., Monographs and Textbooks in Pure and Applied Mathematics 228, in: Difference Equations and Inequalities. Theory, Methods, and Applications, 2. ed. (2000) [2] DOI: 10.1016/j.amc.2005.10.024 · Zbl 1100.39002 [3] Amleh A.M., Int. J. Difference Equ. 3 pp 1– (2008) [4] Amleh A.M., Int. J. Difference Equ. 3 pp 195– (2008) [5] Andruch-Sobilo A., Opuscula Math. 26 pp 387– (2006) [6] A. Andruch-Sobilo and M. Migda, On the rational recursive sequence, Tatra Mt. Math. Publ. 43 (2009), pp. 1–9 [7] DOI: 10.1142/S0218127406015027 · Zbl 1141.37310 [8] DOI: 10.1016/j.amc.2003.08.139 · Zbl 1069.39022 [9] Cull P., Difference Equations. From Rabbits to Chaos (2005) · Zbl 1085.39002 [10] Elaydi S., An Introduction to Difference Equations, 3. ed. (2005) · Zbl 1071.39001 [11] Kulenović M.R.S., Dynamics of Second Order Rational Difference Equations (2002) · Zbl 0981.39011 [12] Sedaghat H., Mathematical Modelling: Theory and Applications 15, in: Nonlinear difference equations. Theory with applications to social science models (2003) · Zbl 1020.39007 [13] DOI: 10.1080/10236190802054126 · Zbl 1169.39006 [14] Stević S., Appl. Math. E-Notes 4 pp 80– (2004)
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