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  • Research Article
  • Open Access

Results and Conjectures about Order Lyness' Difference Equation in , with a Particular Study of the Case

Advances in Difference Equations20092009:134749

https://doi.org/10.1155/2009/134749

  • Received: 4 March 2009
  • Accepted: 14 July 2009
  • Published:

Abstract

We study order Lyness' difference equation in , with and the associated dynamical system in . We study its solutions (divergence, permanency, local stability of the equilibrium). We prove some results, about the first three invariant functions and the topological nature of the corresponding invariant sets, about the differential at the equilibrium, about the role of 2-periodic points when is odd, about the nonexistence of some minimal periods, and so forth and discuss some problems, related to the search of common period to all solutions, or to the second and third invariants. We look at the case with new methods using new invariants for the map and state some conjectures on the associated dynamical system in in more general cases.

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation

Publisher note

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Authors’ Affiliations

(1)
Institut de Mathématiques de Jussieu, UPMC-Paris 06, UMR-CNRS 7586, 75251 Paris, France
(2)
Laboratoire Paul Painlevé, USTL Lille 1, UMR-CNRS 8524, 59655 Villeneuve, France
(3)
Equipe d'Analyse fonctionnelle, IMJ, 16 rue Clisson, 75013 Paris, France

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