Summary: We consider a discrete map proposed by M. Kopel
[Chaos Solitons Fractals 7, No. 12, 2031–2048 (1996; Zbl 1080.91541
)] that models a nonlinear Cournot duopoly consisting of a market structure between the two opposite cases of monopoly and competition. The stability of the fixed points of the discrete dynamical system is analyzed. Synchronization of two dynamics parameters of the Cournot duopoly is considered in the computation of stability boundaries formed by parts of codim-1 bifurcation curves. We discover more on the dynamics of the map by computing numerically the critical normal form coefficients of all codim-1 and codim-2 bifurcation points and computing the associated two-parameter codim-1 curves rooted in some codim-2 points. It enables us to compute the stability domains of the low-order iterates of the map. We concentrate in particular on the second, third and fourth iterates and their relation to the period doubling, 1:3 and 1:4 resonant Neimark-Sacker points.