Global behaviour of the Riccati difference equation of order two

Abstract

The second order rational difference equation is associated with a linear third order difference equation in the same way that the first order Riccati equation (c = 0) is associated with a linear second order equation. This association and other features are used to study the global behaviour of solutions. If and then the above equation has a unique positive fixed point that is stable and attracts all orbits with initial points outside a set M of Lebesgue measure zero in the plane. However, within M there is an invariant subset containing periodic orbits of all possible periods.