Integrability and non-integrability of periodic non-autonomous Lyness recurrences∗

This paper studies non-autonomous Lyness type recurrences of the form xn+2 = (an +xn+1)/xn, where {an} is a k-periodic sequence of positive numbers with primitive period k. We show that for the cases k ∈ {1, 2, 3, 6} the behavior of the sequence {xn} is simple (integrable) while for the remaining cases satisfying this behavior can be much more complicated (chaotic). We also show that the cases where k is a multiple of 5 present some different features. 2000 Mathematics Subject Classification: 37C55, 39A11, 39A20