TY - JOUR

T1 - Some properties of the k-dimensional Lyness' map

AU - Cima, Anna

AU - Gasull, Armengol

AU - Mãosa, Víctor

PY - 2008/7/18

Y1 - 2008/7/18

N2 - This paper is devoted to study some properties of the k-dimensional Lyness' map F(x1, ..., xk) = (x2, ..., xk, (a + ∑ki=2xi)/x1). Our main result presents a rational vector field that gives a Lie symmetry for F. This vector field is used, for k ≤ 5, to give information about the nature of the invariant sets under F. When k is odd, we also present a new (as far as we know) first integral for F○F which allows us to deduce in a very simple way several properties of the dynamical system generated by F. In particular for this case we prove that, except on a given codimension one algebraic set, none of the positive initial conditions can be a periodic point of odd period. © 2008 IOP Publishing Ltd.

AB - This paper is devoted to study some properties of the k-dimensional Lyness' map F(x1, ..., xk) = (x2, ..., xk, (a + ∑ki=2xi)/x1). Our main result presents a rational vector field that gives a Lie symmetry for F. This vector field is used, for k ≤ 5, to give information about the nature of the invariant sets under F. When k is odd, we also present a new (as far as we know) first integral for F○F which allows us to deduce in a very simple way several properties of the dynamical system generated by F. In particular for this case we prove that, except on a given codimension one algebraic set, none of the positive initial conditions can be a periodic point of odd period. © 2008 IOP Publishing Ltd.

U2 - https://doi.org/10.1088/1751-8113/41/28/285205

DO - https://doi.org/10.1088/1751-8113/41/28/285205

M3 - Article

VL - 41

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

M1 - 285205

ER -