On the algebraic difference equations un+2 +un = ψ(un+1) in R, related to a family of elliptic quartics in the plane

We study difference equations of the type un+2 + un = ψ(un+1) in R, with invariant curves given by x2y2 +dxy(x +y)+c(x2 +y2)+bxy +a(x +y)−K = 0. This completes the results about “multiplicative” difference equations of the type un+2un = ψ(un+1) obtained in the previous paper. We reduce first these “additive” difference equations to un+2 +un = α+βun+1 1+u2 n+1 .We study specially the case α = 0, |β| 2. Using the parametrization of the above elliptic quartics by Weierstrass’ elliptic functions, we show that the solutions behave somewhat as in the multiplicative case: if β = 0, there is divergence if the starting point (u1,u0) is not the locally stable fixed point (0, 0), and density of periodic initial points and of initial points with dense orbit in the quartic, with “invariant pointwise chaotic behavior.” We show that the period can be every number n 3, depending on β and on the starting point. © 2006 Elsevier Inc. All rights reserved.