Tropical cycles

Abstract

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.