Global behavior of the solutions of the k-lacunary order 2k Lyness' difference equation in , and of other more general equations

Abstract

For and , we study the k-lacunary order 2k Lyness' difference equation . There are k invariant functions, so that the point , if it is not constant, moves when n varies on a m-dimensional manifold of which is homeomorphic to the m-dimensional torus , for some m, , depending on the starting point . The associated dynamical system has the following properties, if : (1) the starting points with dense orbit in the associated manifold are dense in ; (2) the starting points with periodic orbit are dense in ; (3) there is a form of pointwise chaotic behavior; and (4) every number multiple of k and greater than some number is the period of some orbit. If all solutions have the common period 5k, and we find all the 5-periodic solutions. In a last part, we hint how to generalize these results to some equations of form or related to conics, cubics or quartics.