Abstract We investigate the stability of solutions of the Gumowski–Mira equation with a period-two coefficient: y n + 1 = y n b n + y n 2 − y n − 1 , n = 0 , 1 , … , with b n = { α ⩾ 0 for  n = 2 k , β ⩾ 0 for  n = 2 k + 1 , k = 0 , 1 , … , and the initial values y −1 , y 0 are real numbers.

We investigate the global asymptotic behavior of solutions of the system of difference equations,,, where the parameters,,, and are positive numbers and the initial conditions and are arbitrary nonnegative numbers. We obtain some asymptotic results for the positive equilibrium of this system.

We investigate the rate of convergence of solutions of some special cases of the equation , with positive parameters and nonnegative initial conditions. We give precise results about the rate of convergence of the solutions that converge to the equilibrium or period-two solution by using Poincaré's theorem and an improvement of Perron's theorem.