Abstract
In this paper, taking the factor of service level provided by the manufacturers into consideration, a static duopolistic Bertrand game with service factor is studied first, in which these two oligarchs produce differentiated products. A dynamic game model of duopoly Bertrand with boundedly rational is established with using the gradient mechanism. By using numerical simulation tools, there are two paths for the system to drop into chaos, that is, flip bifurcation and Neimark-Sacker bifurcation. The symmetric structures can be found from two-parameter bifurcation diagrams. Saddle-homoclinic bifurcation also can be observed from the evolution process of phase portraits. In addition, the emergence of intermittent chaos implies that the established system has the capability of self-regulating, where PM-I intermittency, PM-III intermittency and crisis-induced intermittency have been studied. With the help of the critical curves, the qualitative changes on the basin of attraction are investigated. At last, it can be found that the values of product differentiation degree and service spillover effect are not the bigger the better. Keeping these two parameters in a relatively small range will be conducive to the long-term stable operation of the two manufacturers.
Complex dynamical behaviors in a Bertrand game with service factor and differentiated products
