Abstract
In this paper, a dynamical two-stage game with R&D competition and joint profit maximization is built. The stability of all the equilibrium points is discussed through Jury condition, and the stability region of the Nash equilibrium point is then given. The influence of the parameters on the system is discussed, and we find that the firm can even benefit from chaos, when it has higher innovation efficiency and higher adjusting speed. And then the coexistence of multiple attractors is studied using basin of attraction. Our research result shows that the coexisting attractors can be observed in the two-parameter bifurcation diagram. At last, the boundary of feasible region, global bifurcations, and formation mechanism of fractal structure of attracting basin are analyzed through critical curves and noninvertible map theory.
Highlights
With the rapid development of economics, the competition among firms has become more and more fierce
In KMZ model, the forms of R&D are extended to total competition, R&D cartel, research joint ventures (RJVs), and RJVs cartel
It has been proved that the boundary equilibrium points are always unstable, and the Nash equilibrium point is stable only when the parameters meet some conditions
Summary
With the rapid development of economics, the competition among firms has become more and more fierce. Based on the research results of [19], Cavalli and Naimzada [20] built a heterogeneous duopoly Cournot game with reduced rationality, isoelastic demand function, and linear total cost They found that the Nash equilibrium point may lose its stability through a flip bifurcation or a Neimark-Sacker bifurcation. A lot of researchers have discussed the complex dynamical behaviors of nonlinear oligopolies from different aspects, such as differentiated goods [21,22,23,24,25], heterogeneous firms [26,27,28,29,30,31], and delayed decisions [32, 33] Another important issue for dynamical economical model is the coexistence of attractors.
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