On the set of periods of the 2-periodic Lyness' equation

Publicació amb motiu de la International Conference on Difference Equations and Applications (July 22-27, 2012, Barcelona, Spain) amb el títol Difference Equations, Discrete Dynamical Systems and Applications We study the periodic solutions of the non-autonomous periodic Lyness' recurrence u = (a + u )/u, where {a} is a cycle with positive values a,b and with positive initial conditions. Among other methodological issues we give an outline of the proof of the following results: (1) If (a, b) ≠ (1, 1), then there exists a value p(a, b) such that for any p > p(a, b) there exist continua of initial conditions giving rise to 2p-periodic sequences. (2) The set of minimal periods arising when (a, b) ∈ (0,∞) and positive initial conditions are considered, contains all the even numbers except 4, 6, 8, 12 and 20. If a ≠ b, then it does not appear any odd period, except 1.