A BIQUADRATIC SYSTEM OF TWO ORDER ONE DIFFERENCE EQUATIONS: PERIODS, CHAOTIC BEHAVIOR OF THE ASSOCIATED DYNAMICAL SYSTEM

Abstract

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.