Dynamic Evolution, Synchronization and Multistability in an Advertising Game Model with Homogeneous Products and Spillover Effect

Advertising competition among firms may give rise to very complex dynamical behaviors. In this article, under the assumption of spillover effects, a two-stage dynamical Cournot game of advertising competition between two firms, which produce homogeneous products, is developed. Then, the local stability of the equilibrium points is discussed, and stability conditions of the equilibrium points are obtained. in order to reveal the complex dynamical behaviors of the model, both analytical and numerical methods are employed. The research results show that the two coordinate axes and the diagonal of the system are invariant manifolds, and one can obtain the dynamical behaviors on the invariant manifolds by analyzing the Logistic map. In addition, it is found that the advertising efforts of these two firms may he synchronized, even if the system is in a chaotic state. Multistability is another topic of focus. It is found that not only two attractors but also three attractors can coexist in the phase space, and contact bifurcation can also occur during the evolution of the attracting basins. Finally, the impact of corporate advertising efforts on corporate profits is analyzed. It is found that too much advertising effort is harmful to the firms, and a firm with low adjustment speed can even earn more when the system is chaotic.