Construction of mappings with attracting cycles (Q1827180)

A real-valued point-map of-the form \[ x_{n+1}=P_m(x_n), \quad n=0,1,\dots,m>2,\;P= \text{ a polynomial}, \tag{1} \] is considered from an inverse point of view. The problem consists of choosing the coefficients of the powers of \(x_n\) so that the iterated polynomial equation \[ x_{n+k}=x_n, \quad k\geq 2 \text{ (fixed in advance)}, \tag{2} \] admits \(k\) distinct real roots (points of a \(k\)-cycle). Various interpolations (Newton, Lagrange, Hermite,\dots) are used to determine the coefficients of \(P_m\) so that (2) admits the postulated roots, although no new method is proposed to actually determine any of them (except numerically). Even for fixed \(m\) and \(k\) the problem has no unique solution, because specifying one singularity does not define the ``general solution'' of the discrete problem (1). For \(m=2\) its main features were determined already by Myrberg long ago. The motive of the authors seems to be an eventual extension of \textit{T.-Y. Li} and \textit{J.A. Yorke}'s claim [Am. Math. Mon. 82, 985--992 (1975; Zbl 0351.92021)]: Period three implies ``chaos''! - The bibliography consists of only four publications, one of which is still to appear. In an applied context the `contribution' is irrelevant, because it sheds no light on the design, construction and operation of frequency dividers. (They are based on synchronized subharmonics, corresponding to cycles in the Poincaré representation.)