No abstract

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.

We study the existence of periodic solutions of the nonautonomous periodic Lyness' recurrence u n+2 = (a n + u n+1 )/u n , where {a n } n is a cycle with positive values a, b and with positive initial conditions. It is known that for a = b = 1 all the sequences generated by this recurrence are 5-periodic. We prove that for each pair (a, b) = (1, 1) there are infinitely many initial conditions giving rise to periodic sequences, and that the family of recurrences have almost all the even periods. If a = b, then any odd period, except 1, appears.

We study the "homographic" system of order one difference equations in

For a . 0 and u 0 ; u 1 ; . . . ; u 2k21 . 0, we study the k-lacunary order 2k Lyness' difference equation u n u n22k ¼ u n2k þ a. There are k invariant functions, so that the point M n ¼ ðu nþ2k21 ; u nþ2k22 ; . . . ; u n Þ, if it is not constant, moves when n varies on a m-dimensional manifold SðM 0 Þ of R þ2k * which is homeomorphic to the m-dimensional torus T m , for some m, 1 # m # k, depending on the starting point M 0 . The associated dynamical system F a ; R þ2k * À Á has the following properties, if a -1: (1) the starting points M 0 with dense orbit in the associated manifold SðM 0 Þ are dense in R þ2k * ;(2) the starting points with periodic orbit are dense in R þ2k * ; (3) there is a form of pointwise chaotic behavior; and (4) every number multiple of k and greater than some number kN a is the period of some orbit. If a ¼ 1 all solutions have the common period 5k, and we find all the 5-periodic solutions. In a last part, we hint how to generalize these results to some equations of form u nþ2k u n ¼ cðu nþk Þ or u nþ2k þ u n ¼ cðu nþk Þ related to conics, cubics or quartics.

We study in [Formula: see text] the biquadratic system of two order one difference equations [Formula: see text] for some values of the parameters. We show that there is an invariant function G, and so that the orbit of a point (u0, v0) in some invariant open set U is on an invariant ellipse, and that the restriction on this ellipse of the associated dynamical system is conjugated to a rotation on a circle. The equilibrium is locally stable and the solutions (un, vn) are permanent. We show also that the starting points with periodic orbit are dense in U, and that every integer p ≥ N(a, b, c) is the minimal period of a periodic solution (un, vn). Moreover, the restriction of the dynamical system to the invariant compact "annulus" {K1 ≤ G ≤ K2} has global sensitivity to initial conditions, for inf U G < K1 < K2 < sup U G. Otherwise, outside U the solutions tend to infinity. At last we prove that the possible rational periodic solutions, when a, b, c are rational, may only be two or three-periodic, and we determine exactly the triples (a, b, c) for which such rational two or three-periodic solutions exist.

We prove that the level curves of some differentiable functions of two variables with unique critical point are diffeomorphic to the circle T, and show how this result can be used in the study of local stability of dynamical systems in dimension 2 with invariant function, without using the Hessian. We extend the results to the level sets of an invariant function of dynamical systems, with a synthesis exposition of examples of improvements of previously studied order q difference equations with invariant. In fine we present some differential tools for the study of the topological nature of invariant level sets in dimension at least three.