Global behavior of the higher order rational Riccati difference equation

摘要

Let k be a positive integer and a0,a1,…,ak be non-negative real numbers with ak>0. We show that if gcd{i;ai-1>0,1⩽i⩽k+1}=1 then the rational Riccati difference equation of order kxn+1=a0+a1xn+a2xnxn-1+⋯+akxnxn-1⋯xn-k+1,n=0,1,2,…has a unique positive equilibrium point that is stable and attracts all solutions with initial points outside a set of zero Lebesgue measure. This holds in particular if a0+ak-1>0. The case k=3 is studied in detail.