T. Wanner We investigate global dynamics of the equation xn+1=xn−12bxnxn−1+cxn−12+f,n=0,1,2,…, where the parameters b,c, and f are nonnegative numbers with condition b + c > 0,f ≠ 0 and the initial conditions x−1,x0 are arbitrary nonnegative numbers such that x−1+x0>0. We obtain precise characterization of basins of attraction of all attractors of this equation and describe the dynamics in terms of bifurcations of period‐two solutions. Copyright © 2015 John Wiley & Sons, Ltd.

By using the Kolmogorov-Arnold-Moser (KAM) theory, we investigate the stability of the positive elliptic equilibrium point of the difference equation xn+1 = Ax3n +B axn−1 , n = 0, 1, 2, . . . where the parameters A,B, a and the initial conditions x −1, x0 are positive numbers. The specific feature of this difference equation is the fact that we were not able to use the invariant to prove stability or to find feasible periods of the solutions.

By using the KAM theory, we investigate the stability of the equilibrium solution of a certain difference equation. We also use the symmetries to find effectively the periodic solutions with feasible periods. The specific feature of this difference equation is the fact that we were not able to use the invariant to prove stability or to find feasible periods of the solutions.

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.

C Dedicated to Professor Mustafa Kulenovic on the occasion of his 60th birthday Abstract. We investigate the global asymptotic behavior of solutions of the following anti-competitive system of difference equations xn+1 = 1yn A1 + xn ; yn +1 = 2xn + 2yn yn ; n = 0; 1;:::; where the parameters 1; 2; 2;A1 are positive numbers and the initial conditions x0 0;y0 > 0. We find the basins of attraction of all at- tractors of the system, which are the equilibrium point and period-two solutions.

We first investigate the Lyapunov stability of the period-three solution of Todd's equation with a period-three coefficient: where α,β, and γ positive. Then for k = 2,3,… we extend our stability result to the k-order equation, where pn is a periodic coefficient of period k with positive real values and x-k+1,…,x-1, x0 ∈ (0, ∞).