We investigate global dynamics of the following second order rational difference equation x n + 1 = x n x n − 1 + α x n + β x n − 1 a x n x n − 1 + b x n − 1 , where the parameters α , β , a , b are positive real numbers and initial conditions x − 1 and x 0 are arbitrary positive real numbers. The map associated to the right-hand side of this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the corresponding parametric space. In most cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability.

We investigate the global asymptotic stability and Naimark-Sacker bifurcation of the dierence equation xn+1 = F bxnxn 1 +cx 2 1 +f ; n = 0; 1;:::

By using the KAM theory we investigate the stability of equilibrium solutions of the Gumowski-Mira equation: x n+1 = (2ax n)/(1 + x n 2) − x n−1, n = 0,1,…, where x −1, x 0 ∈ (−∞, ∞), and we obtain the Birkhoff normal forms for this equation for different equilibrium solutions.

We investigate the global asymptotic behavior of solutions of the following anti-competitive system of rational difference equations: xn+1=γ1ynA1+xn,yn+1=β2xnA2+B2xn+yn,n=0,1,…, where the parameters γ1, β2, A1, A2 and B2 are positive numbers and the initial conditions (x0,y0) are arbitrary nonnegative numbers. We find the basins of attraction of all attractors of this system, which are the equilibrium points and period-two solutions. MSC:39A10, 39A11.

AbstractWe investigate the global asymptotic behavior of solutions of the following anti-competitive system of rational difference equations: xn+1=γ1ynA1+xn,yn+1=β2xnA2+B2xn+yn,n=0,1,…, where the parameters γ1, β2, A1, A2 and B2 are positive numbers and the initial conditions (x0,y0) are arbitrary nonnegative numbers. We find the basins of attraction of all attractors of this system, which are the equilibrium points and period-two solutions.MSC:39A10, 39A11.