摘要：This paper studies the difference equation x n + 3 x n = a + x n + 1 + x n + 2 + γ x n 2 , where a and γ are arbitrary positive real numbers and the initial values x 0 , x 1 , x 2 > 0 . It is known that for γ = 0 the above equation is the third-order Lyness’ one, studied in several papers. Using an extension of the quasi-Lyapunov method, we prove that for 0 < γ < 1 the sequences generated by the perturbed third-order Lyness equation are globally asymptotically stable. Moreover, we show that if γ ≥ 1 all solutions of it converge to +∞. Therefore, the values 0 and 1 are two bifurcation points for the equation containing the parameter γ. MSC:39A11, 39A20.
关键词：difference equation ; quadratic perturbations ; bifurcation point ; first integral ; Lyapunov function ; global asymptotic stability
