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Dynamics and behavior of a second order rational difference equation. (English) Zbl 1291.39026
The object of study of the paper is the rational difference equation $\displaystyle{x_{n+1} = ax_n + {{b+cx_{n-1}}\over{d+ex_{n-1}}}\;,\;n=0,1,2,\ldots}$ with $$a>0$$, $$b>0$$, $$c>0$$, $$d>0$$, $$e>0$$ and the initial conditions subject to $$x_{-1}>0$$, $$x_0>0$$. The following results are proved
i) the unique positive equilibrium is the positive root of $e(1-a)\bar{x}^2 + (d-da-c)\bar{x} - b = 0$ and is asymptotically stable by the first approximation if $\displaystyle{(d+e\bar{x})^2>{{|cd-be|}\over{1-a}}\;,\;a<1}$ ii) every solution is bounded provided $$a<1$$ and unbounded if $$a>1$$. iii) the equation has positive prime period 2 solutions if and only if $(c-ad-d)^2(1+a) + 4[be+(c-ad-d)ad]>0$ iv) The positive equilibrium $$\bar{x}$$ is a global attractor if either $cd\geq be\;,\;c>d(1-a)\;,\;a<1$ or $cd\leq be\;,\;a<1$ The paper is illustrated by some numerical examples.

MSC:
 39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type 39A22 Growth, boundedness, comparison of solutions to difference equations 39A23 Periodic solutions of difference equations 39A30 Stability theory for difference equations