In this paper, we investigate the Cournot-Bertrand double oligopoly hybrid competition model proposed by Zhu et al. (2021). By using the central manifold reduction theorem we analyze the structural stability of the model for three fixed points. We show that a subcritical (supercritical) flip bifurcation occurs at first (second and third) fixed point. This reflects that in the market competition, there will be a 2-period cycle between two firms. When subcritical flip bifurcation occurs, the 2-period cycle is unstable and the objective function values of both firms gradually deviates from the cycle. On the contrary, when the supercritical bifurcation occurs, the 2-period cycle is stable, and the objective function values of both firms gradually tend to the stable cycle. We also prove that a transcritical bifurcation will occur at the second fixed point, which indicates that the two firms will present two equilibrium states on both sides of the fixed point in the small neighborhood, but the stability of these two equilibrium states is opposite.