Qualitative properties and bifurcations of a Cournot-Bertrand duopoly mixed competition model

In this paper, the qualitative properties of the fixed points in the non-hyperbolic cases, codimension-one bifurcations and weak resonances of a Cournot-Bertrand duopoly mixed competition model are explored. The two firms adopt different decision variables and different objective functions, which are more consistent with the actual economic market situation. The qualitative properties of all the fixed points in the non-hyperbolic cases are investigated using the reduction principle and the center manifold theorem. After that, all the potential codimension-one bifurcations, including transcritical bifurcation, supercritical or subcritical flip bifurcation and Neimark–Sacker bifurcation are analyzed using the bifurcation theory and the center manifold theorem. The direction, stability, and even the explicit approximate expression are derived for each type of bifurcation. By perturbing the closed invariant curve caused by the Neimark–Sacker bifurcation, the 2 : 5 weak resonance associated with Arnold's tongue is theoretically proved, and the absence of 1 : 6 and 5 : 6 weak resonances is further analyzed. A large number of numerical simulations show complete consistency with all theoretical analyses. Moreover, the continuation method is used to conduct numerical bifurcation analyses, further verifying the correctness of theoretical analyses, and testing more codimension 2 bifurcations, such as fold-flip bifurcation, generalized flip bifurcation and 1 : 2 strong resonance. In addition, the economic implications of these bifurcations are also explained accordingly.