Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let k   be a positive integer and a0,a1,…,aka0,a1,…,ak be non-negative real numbers with ak>0ak>0.  show that if gcd{i;ai-1>0,1⩽i⩽k+1}=1 then the rational Riccati difference equation of order k#R##N##R##N#xn+1=a0+a1xn+a2xnxn-1+⋯+akxnxn-1⋯xn-k+1,n=0,1,2,…#R##N##R##N#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>0a0+ak-1>0. The case k=3k=3 is studied in detail.
['Azizi Raouf']
I am writing a research paper on a idea Global behavior of the higher order rational Riccati difference equation
[{"from": "human", "value": "I am writing a research paper on a idea Global behavior of the higher order rational Riccati difference equation"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let k \u00a0 be a positive integer and a0,a1,\u2026,aka0,a1,\u2026,ak be non-negative real numbers with ak>0ak>0.  show that if gcd{i;ai-1>0,1\u2a7di\u2a7dk+1}=1 then the rational Riccati difference equation of order k#R##N##R##N#xn+1=a0+a1xn+a2xnxn-1+\u22ef+akxnxn-1\u22efxn-k+1,n=0,1,2,\u2026#R##N##R##N#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>0a0+ak-1>0. The case k=3k=3 is studied in detail."}]