On 2- and 3-periodic Lyness difference equations

摘要

We describe the sequences {x _n} _n given by the non-autonomous second-order Lyness difference equations x _(n+2) = (a _n + x _(n+1))/x ＿n, where {a _n} _n is either a 2-periodic or a 3-periodic sequence of positive values and the initial conditions x _1x _2 are also positive. We also show an interesting phenomenon of the discrete dynamical systems associated with some of these difference equations: the existence of one oscillation of their associated rotation number functions. This behaviour does not appear for the autonomous Lyness difference equations.