Bifurcation, intermittent chaos and multi-stability in a two-stage Cournot game with R&D spillover and product differentiation

Abstract

In this paper, a dynamical two-stage Cournot duopoly game with R&D spillover effect and product differentiation is established. The stability of all the equilibrium points is studied using Jury criterion, and then the stability condition is given. The direction of flip bifurcation is given by using central manifold theorem and norm form theory. By numerical simulation, two routes to chaos are studied through 2-D bifurcation diagram, one of which is flip bifurcation and the other is Neimark-Sacker bifurcation at period-2 point. The system will lose its stability as the speed of adjustment increases. In addition, a common nonlinear phenomenon named intermittent chaos is observed in the built model. Two types of intermittent chaos are displayed in time series plots and phase diagrams, one of which is PM-III intermittent chaos and the other is induced by crisis. And then four types of coexistence of attractors are illustrated through basin of attraction, which are coexistence of periodic attractor and chaotic attractor, coexistence of multiple chaotic attractors, coexistence of periodic attractor, quasi-periodic attractor and chaotic attractor, and coexistence of multiple periodic attractors, respectively.