Abstract
By using the Kolmogorov-Arnold-Moser (KAM) theory, we investigate the stability of two elliptic equilibrium points (zero equilibrium and negative equilibrium) of the difference equation tn+1=alpha tn+beta tn2-tn-1,n=0,1,2, horizontal ellipsis , where are t-1, t0, alpha is an element of R, alpha not equal 0, beta>0. By using the symmetries we find the periodic solutions with some periods. Finally, some numerical examples are given to verify our theoretical results.