On the algebraic difference equations [formula omitted] in [formula omitted], related to a family of elliptic quartics in the plane

Abstract

We study difference equations of the type u n + 2 + u n = ψ ( u n + 1 ) in R , with invariant curves given by x 2 y 2 + d x y ( x + y ) + c ( x 2 + y 2 ) + b x y + a ( x + y ) − K = 0 . This completes the results about “multiplicative” difference equations of the type u n + 2 u n = ψ ( u n + 1 ) obtained in the previous paper. We reduce first these “additive” difference equations to u n + 2 + u n = α + β u n + 1 1 + u n + 1 2 . 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 ( 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 β and on the starting point.