L.A.S. and negative Schwarzian derivative do not imply G.A.S. in Clark's equation (Q300355)

The paper deals with stability problems for the equilibrium of the Clark difference equation \[ x_{n+1} = \alpha x_n + (1-\alpha)h(x_{n-k}) \] with \(h:I\mapsto I\) being unimodal or decreasing with negative Schwarzian derivative defined by \[ \displaystyle{(Sh)(z) = {{h'''(z)}\over{h'(z)}} - {{3}\over{2}}\left( {{h''(z)}\over{h'(z)}}\right)^2}. \] The paper introduces the so-called class \(S\) of functions of real variable (not to mistake for the \(S\) (schlicht) class of functions of complex variable from the Bieberbach conjecture). It is proved via the Neimark-Sacker bifurcation theory that ``the conjecture of local asymptotic stability (L.A.S.) (stability by the first approximation) implying global asymptotic stability (G.A.S.) of the equilibrium'' fails for \(k\geq 3\) -- some sui-generis delay dependent global asymptotic stability.