Advances in Difference Equations
Volume 2005 (2005), Issue 3, Pages 227-261
On the algebraic difference equations in , related to a family of elliptic quartics in the plane
1Institut Mathématique de Jussieu, University Paris 6 and CNRS, Paris, France
2Equipe d'Analyse Fonctionnelle, Institut Mathématique de Jussieu, 16 rue Clisson, Paris 75013, France
3Laboratoire Paul Painlevé, University Lille 1 and CNRS, Lille, France
Received 20 October 2004; Revised 27 January 2005
Copyright © 2005 G. Bastien and M. Rogalski. 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.
We continue the study of algebraic difference equations of the type
, which started in a previous paper.
Here we study the case where the algebraic curves related to the
equations are quartics of the plane. We prove, as in “on
some algebraic difference equations in , related to families of conics or cubics: generalization of the
Lyness' sequences” (2004), that the solutions are persistent and bounded, move on the
positive component of the quartic which passes through , and diverge if is not the equilibrium, which is locally stable. In fact, we study the
dynamical system , , , , in , and show that its restriction to is conjugated to a rotation on the circle. We give the possible
periods of solutions, and study their global behavior, such as the density of initial
periodic points, the density of trajectories in some curves, and a
form of sensitivity to initial conditions. We prove a dichotomy
between a form of pointwise chaotic behavior and the existence of
a common minimal period to all nonconstant orbits of .