On the rational recursive sequence \(x_{n+1}= \frac {bx_n^2} {1+x_{n-1}^2}\) (Q5939119)

For the recursive sequence \(x_{n+1}= \frac{bx_n^2} {1+x_{n-1}^2}\) with \(b\) and the initial terms \(x_{-1}\), \(x_0\) being positive real numbers, \textit{V. L. Kocic} and \textit{G. Ladas} [Global behavior of nonlinear difference equations of higher order with applications. Mathematics and its Applications (Dordrecht). 256. Dordrecht: Kluwer Academic Publisher (1993; Zbl 0787.39001)] showed that if \(b=2\) then every such sequence converges to 0 or 1. In this paper the authors establish the following results: If \(b<2\) then each sequence above converges to 0; If \(b>2\) then one of the following holds: (i) it converges to 0; (ii) it converges to one of the roots of \(x^2-bx+1=0\), or (iii) it is strictly oscillatory about the greater root of the equation in (ii). Then they ask if the last two cases really occur except for when every term of the sequence is a constant, i.e., one of the roots of the equation \(x^2-bx+1=0\).