By using the KAMtheory we investigate the stability of equilibrium solutions of the Gumowski-Mira equation: xn+1 = (2axn)/(1 + x2n) – xn-1, n = 0, 1, …, where x-1, x0, ∈ (−∞,∞), and we obtain the Birkhoff normal forms for this equation for different equilibrium solutions
