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

Abstract

This paper studies non-autonomous Lyness-type recurrences of the form x _n+2 = (a _n + x _n+1)/x _n , where {a _n } is a k-periodic sequence of positive numbers with primitive period k. We show that for the cases k ∈ {1, 2, 3, 6}, the behaviour of the sequence {x _n } is simple (integrable), while for the remaining cases satisfying this behaviour can be much more complicated (chaotic). We also show that the cases where k is a multiple of 5 present some different features.

Keywords:

• Integrability and non-integrability of discrete systems
• numerical chaos
• periodic difference equations
• QRT maps
• rational and meromorphic first integrals

Acknowledgements

We thank Joan Carles Tatjer, for his suggestion to introduce the maps G _[k] for a better approach to the numerical study of the maps F _[k], and John A.G. Roberts, who gave us very useful information about the interest of classical Lyness recurrences in the context of discrete integrability.

GSD-UAB and CoDALab Groups are supported by the Government of Catalonia through the SGR program. They are also supported by Ministry of Economy and Competitiveness of the Spanish Government through grants MTM2008-03437 (first and second authors), DPI2008-06699-C02-02 and DPI2011-25822 (third author).