On the Algebraic Difference Equations

We continue the study of algebraic difference equations of the type un+2un = ψ(un+1), which started in a previous paper. Here we study the case where the algebraic curves related to the equations are quartics Q(K) of the plane. We prove, as in “on some algebraic difference equations un+2un = ψ(un+1) in R∗, related to families of conics or cubics: generalization of the Lyness’ sequences” (2004), that the solutions Mn = (un+1,un) are persistent and bounded, move on the positive component Q0(K) of the quartic Q(K) which passes through M0, and diverge if M0 is not the equilibrium, which is locally stable. In fact, we study the dynamical system F(x, y) = ((a+ bx + cx2)/y(c + dx + x2),x), (a,b,c,d)∈R+, a+ b > 0, b+ c+ d > 0, in R∗, and show that its restriction to Q0(K) is conjugated to a rotation on the circle. We give the possible periods of solutions, and study their global behavior, such as the density of initial periodic points, the density of trajectories in some curves, and a form of sensitivity to initial conditions. We prove a dichotomy between a form of pointwise chaotic behavior and the existence of a common minimal period to all nonconstant orbits of F.