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We study the homographic system u(n+1)u(n)=c+d/v(n), v(n+1)v(n)=c+d/u(n+1) inthe positive quadrant. The orbit of a point is contained in an invariant cubiccurve, and the restriction to the positive part of this cubic of the associateddynamical system is conjugated to a rotation on the circle. For a denseinvariant set of initial points the solutions are periodic, and if c=1 (this isalways possible) every integer n\geq N(d) is the minimal period of someperiodic solution. Every n\geq 11 is the minimal period of some solution forsome d>0, and we find exactly the set of such minimal periods between 2 and 10.The associated dynamical system has sensitiveness to initial conditions onevery compact set not containing the fixed point.

A QRT-system of two order one homographic difference equations:  conjugation to rotations, periods of periodic solutions, sensitiveness to  initial conditions