On periodic solutions of 2-periodic Lyness' equations

Abstract

We study the existence of periodic solutions of the nonautonomous periodic Lyness' recurrenceun+2 = (an + un+1)/un, where {an}n is a cycle with positive values a, b and with positive initial conditions. It is known that for a = b = 1 all the sequences generated by this recurrence are 5-periodic. We prove that for each pair (a, b) ≠ (1, 1) there are infinitely many initial conditions giving rise to periodic sequences, and that the family of recurrences have almost all the even periods. If a ≠ b, then any odd period, except 1, appears.