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

TL;DR

This paper analyzes a specific homographic difference system, showing it is conjugate to a circle rotation, characterizing periodic solutions, their periods, and sensitivity to initial conditions within the positive quadrant.

Contribution

It establishes the conjugation of the system to a rotation on the circle and characterizes the periods of solutions and their dependence on initial conditions.

Findings

Solutions are contained in an invariant cubic curve.

Periodic solutions exist with minimal periods for all integers n ≥ 11.

The system exhibits sensitivity to initial conditions outside the fixed point.

Abstract

We study the homographic system u(n+1)u(n)=c+d/v(n), v(n+1)v(n)=c+d/u(n+1) in the positive quadrant. The orbit of a point is contained in an invariant cubic curve, and the restriction to the positive part of this cubic of the associated dynamical system is conjugated to a rotation on the circle. For a dense invariant set of initial points the solutions are periodic, and if c=1 (this is always possible) every integer n\geq N(d) is the minimal period of some periodic solution. Every n\geq 11 is the minimal period of some solution for some 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 on every compact set not containing the fixed point.