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

Anna Cima, Armengol Gasull, Víctor Mañosa

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5 Citations (Scopus)


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. © 2013 Copyright Taylor and Francis Group, LLC.
Original languageEnglish
Pages (from-to)518-538
JournalDynamical Systems
Issue number4
Publication statusPublished - 1 Dec 2013


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


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