We study the Lyness' difference equation in for and initial conditions u 1, We prove a conjecture (in the reference of Barbeau, E, Gelbord, B, Tanny, S. (1995) “Periodicities of solutions of the generalized Lyness recursion”, Journal of Difference Equations and Applications, 1, pp. 291–306) and show if that the periodic points (u 1, u 0) are dense in and that the set of points (u 1, u 0) with dense orbit in the bounded component of the associated Lyness' cubic is also dense in Moreover, all points of have sensitivity to initial conditions (L is the fixed point). We prove that for a given there exist arbitrary long periods. We determinate the integers which are period of some Lyness' sequence, and research periodic rational solutions of Lyness' equation. Our methods will be generalized for studying some algebraic difference equations of the form related to families of conics, cubics or quartics [in the reference of Bastien and Rogalski, “On some algebraic difference equations in related to families of conics or cubics: generalization of Lyness' sequences”, to appear in Journal of Mathematical Analysis and Applications]. mro@ccr.jussieu.fr