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We continue the study of algebraic difference equations of the type u n+2 u n = ψ(u n+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 u n+2 u n = ψ(u n+1 ) in R + * , related to families of conics or cubics: generalization of the Lyness' sequences" (2004), that the solutions M n = (u n+1 ,u n ) are persistent and bounded, move on the positive component Q 0 (K) of the quartic Q(K) which passes through M 0 , and diverge if M 0 is not the equilibrium, which is locally stable. In fact, we study the dynamical system F(x, y) = ((a+ bx + cx 2 )/y(c + dx, and show that its restriction to Q 0 (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.

We study order q Lyness' difference equation in R * : u n q u n a u n q−1 · · · u n 1 , with a > 0 and the associated dynamical system F a in R q * . We study its solutions divergence, permanency, local stability of the equilibrium . We prove some results, about the first three invariant functions and the topological nature of the corresponding invariant sets, about the differential at the equilibrium, about the role of 2-periodic points when q is odd, about the nonexistence of some minimal periods, and so forth and discuss some problems, related to the search of common period to all solutions, or to the second and third invariants. We look at the case q 3 with new methods using new invariants for the map F 2 a and state some conjectures on the associated dynamical system in R q * in more general cases.

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In this paper, we present the current teaching of linear algebra in first year of French science university and the main difficulties of the students. Then we give a brief