Open Access

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

Advances in Difference Equations20092009:134749

Received: 4 March 2009

Accepted: 14 July 2009

Published: 26 August 2009


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.

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Authors’ Affiliations

Institut de Mathématiques de Jussieu, Paris, France
Laboratoire Paul Painlevé, Villeneuve, France
Equipe d'Analyse fonctionnelle, IMJ, Paris, France


© G. Bastien and M. Rogalski. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.