On some algebraic difference equations un+2un = ψ(un+1) in R+∗ , related to families of conics or cubics: generalization of the Lyness’ sequences

In this paper and in a forthcoming one, we study difference equations in R+∗ of the types un+2un = a + bun+1 + u2 n+1, (1) un+2un = a + bun+1 + cu2 n+1 c + un+1 , (2) un+2un = a + bun+1 + cu2 n+1 c +dun+1 + u2 n+1 , (3) which are linked to families of conics, cubics and quartics, respectively. These equations generalize Lyness’ one un+2un = a + un+1 studied in several papers, whose behavior was completely elucidated in [G. Bastien, M. Rogalski, in press] through methods which are transposed in the present paper for the study of (1) and (2), and in the forthcoming one for (3). In particular we prove in the present paper a form of chaotic behavior for solutions of difference equations (1) and (2), and find all the possible periods for these solutions.