Global dynamics, multistability and synchronization in a cross–ownership oligopoly model

This paper studies a Cournot duopoly model with cross–ownership under boundedly rational output adjustment, where ownership links between firms couple firm payoffs and make the dynamics jointly depend on competition intensity, ownership structure, and adjustment behavior. Because the system is noninvertible, the global dynamics are examined through the structure of the feasible region, the basins of attraction, and their boundaries. The feasible boundary is determined not only by invariant segments on the coordinate axes, but also by the interaction between boundary preimages and critical curves. As the cross–ownership parameters vary, tangencies and intersections between critical curves and the feasible boundary allow escape regions to enter the feasible region as internal holes. Successive preimages transform the basin of attraction from a simply connected set into a structure with holes and layered boundaries, eventually leading to attractor destruction after contact with the basin boundary. Changes in cross–ownership proportions also produce multistability, including attractor merging and disappearance associated with unstable periodic orbits. Moreover, under symmetric conditions we analyze synchronization. Cross–ownership affects both the stability range of the competitive equilibrium and the transverse stability of the synchronization manifold, thereby determining whether the outputs of the two firms evolve synchronously.