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

We study difference equations of the type u(n+2) + u(n) = psi(u(n+1)) in R, with invariant curves given by x(2)y(2) + dxy(x + y) + c(x(2) + Y-2) + bxy + a(x + y) - K = 0. This completes the results about "multiplicative" difference equations of the type u(n+2)u(n) = psi(u(n+1)) obtained in the previous paper. We reduce first these "additive" difference equations to u(n+2) +u(n) = alpha+beta u(n+1)/1+u(n+1)(2). We study specially the case alpha = 0, vertical bar beta vertical bar <= 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 beta not equal 0, there is divergence if the starting point (u(1), u(0)) 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 beta and on the starting point. (c) 2006 Elsevier Inc. All rights reserved.