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Global behaviours of rational difference equations of orders two and three with quadratic terms. (English) Zbl 1169.39006
Global behavior of solutions of rational difference equations
\[ x_{n+1}=\frac{ax_{n-1}}{x_{n}x_{n-1}+b},\quad n=0,1,2,\dots, \]
and
\[ x_{n+1}=\frac{ax_{n}x_{n-1}}{x_{n}+bx_{n-2}},\quad n=0,1,2,\dots \]
where \(a,b>0\) and \(x_{-1},x_{0}\) are real, are derived. These equations are related to each other via semiconjugate relations that also let us reduce them to first order equations. Using this approach, the ‘forbidden sets’ (so that if the initial values are chosen outside them, then the corresponding solution will never enter these sets) of each equation are determined explicitly. Furthermore, for initial values outside the forbidden sets, the corresponding solutions may converge to \(0,\) or to a positive fixed point, or may be periodic of period \(2,\) or unbounded. In some cases, different types of solutions coexist depending on the initial values.

MSC:
39A22 Growth, boundedness, comparison of solutions to difference equations
39A23 Periodic solutions of difference equations
39A10 Additive difference equations
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References:
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