The rational recursive sequence \(x_{n+1}=(\beta x^ 2_ n)/(1+x^ 2_{n-1})\)
The authors consider the global behavior of the rational sequence satisfying the equation (1) \(x_{n+1}=(\beta x^ 2_ n)/(1 + x^ 2_{n-1})\), \(n \in Z_ + :=\{0,1,2, \dots\}\), where \(\beta \in (0,\infty)\) is a parameter, and the initial values \(x_{-1}\), \(x_ 0\) are arbitrary positive numbers. The results obtained can be summarized as follows: A. Every solution of equation (1) with \(\beta \in (0,\infty)\) is bounded; B. When \(0<\beta<2\), the unique equilibrium point \(\overline x_ 1=0\) of (1) is globally asymptotically stable; C. When \(\beta=2\), equation (1) has two equilibria \(\overline x_ 1=0\) and \(\overline x_ 2=1\), where \(\overline x_ 1=0\) is locally asymptotically stable and \(\overline x_ 2=1\) is unstable. There exists a solution of (1) which is strictly increasing to \(\overline x_ 2=1\). Any solution of (1) which is neither identically equal to one nor strictly increasing to 1 obeys \(x_ n \to 0\) as \(n \to \infty\); D. When \(\beta>2\), equation (1) has three equilibria, namely, \(\overline x_ 1=0\), \(\overline x_ 2={1 \over 2} (\beta-\sqrt {\beta^ 2-4})\) and \(\overline x_ 3={1 \over 2} (\beta + \sqrt {\beta^ 2-4})\), where \(\overline x_ 1\) is locally asymptotically stable, \(\overline x_ 2\) is a saddle point, and \(\overline x_ 3\) is a repellor (hence they are all unstable). There exist two solutions of equation (1) such that one of them is strictly increasing to the smallest positive equilibrium \(\overline x_ 2\) and the other obeys \(x_{n-1} > \overline x_ 2\) for all \(n \in Z_ +\). The method of proof used is quite elementary.
scientific article; zbMATH DE number 641429
scientific article