Dynamics of the third order Lyness' difference equation

Anna Cima, Armengol Gasull, Víctor Mañosa

Research output: Contribution to journalReview articleResearchpeer-review

20 Citations (Scopus)

Abstract

This paper studies the iterates of the third order Lyness' recurrence [image omitted], with positive initial conditions, being a also a positive parameter. It is known that for a=1 all the sequences generated by this recurrence are 8-periodic. We prove that for each [image omitted] there are infinitely many initial conditions giving rise to periodic sequences which have almost all the even periods and that for a full measure set of initial conditions the sequences generated by the recurrence are dense in either one or two disjoint bounded intervals of . Finally, we show that the set of initial conditions giving rise to periodic sequences of odd period is contained in a co-dimension one algebraic variety (so it has zero measure) and that for an open set of values of a it also contains all the odd numbers, except finitely many of them.
Original languageEnglish
Pages (from-to)855-884
JournalJournal of Difference Equations and Applications
Volume13
Issue number10
DOIs
Publication statusPublished - 1 Oct 2007

Keywords

  • Circle map
  • Difference equation
  • Discrete dynamical system
  • First integral
  • Periodic orbit
  • Rotation number

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