Dynamic behavior of a recursive sequence
The authors study the behavior of solutions of the difference equation (\(*\)) \(x_{n+1}=\alpha-x_{n-1}/x_n\) (\(n=0,1,2,\dots\)), where \(\alpha\in{\mathbb R}\) is a given number. According to the value of \(\alpha\) and the initial conditions, the authors show that the negative equilibrium point \(\bar x:=\alpha-1\) is globally asymptotically stable (for \(\alpha<-1\)), or every negative solution converges to a two-cycle (for \(\alpha=-1\)), or (\(*\)) has unbounded solutions (for \(-1<\alpha\leq1\)), or (\(*\)) is chaotic (for \(1<\alpha<3\)), or the positive equilibrium point \(\bar x=\alpha-1\) is a global attractor with a restricted basin (for \(\alpha>3\)). Similar results were recently obtained for the difference equation (\(*\)) but with the plus instead of the minus sign.
scientific article; zbMATH DE number 2117758