The Lyness equation
(1)
\begin{equation}{X_{n + 1}} = \frac{{{X_n} + a}}{{{X_{n - 1}}}},\,(a,{x_1},{x_2} > 0)\end{equation}
was introduced in 1947 by Lyness [1] and it, and related equations, have long been studied; see [1, 2, 3, 4, 5, 6, 7] and references therein. Perhaps surprisingly, all solutions of (1) are bounded (i.e. for all
x1,
x2, the set {
xn} is bounded) - we will show that below. Furhter, there often exist periodic solutions (i.e.
xn =
xn+N for all
n in which case we say that (
xn) has period
N). See [8] for a discussion of which periods are possible for a given α. We note that a sequence of period, say, 5 also has periods 10, 15, 20, …. so we use the term
minimal period for the smallest positive
N such that
xn =
xn+N for all
n.