* Correspondence: lds1010@sina. com School of Economics and Management, University of South China, Hengyang, Hunan 421001, People’s Republic of China Full list of author information is available at the end of the article Abstract In this paper, we consider the rule of trajectory structure for a kind of second-order rational difference equation. With the change of the initial values, we find the...

We describe the asymptotic behaviour and the stability properties of the solutions to the nonlinear second order difference equation xn+1 = xn−1 a + bxnxn−1 , n ≥ 0, for all values of the real parameters a, b, and any initial condition (x−1, x0) ∈ R .

In this paper, the global behavior for a class of nonlinear difference equation is considered, mainly for the global asymptotical stability of zero equilibrium with attractive basin, the existence of unbounded solutions, the existence of period two solutions, and the existence of oscillatory solutions, etc. Our results extend and generalize the known ones.