Abstract
•Knowledge spillover effect is introduced into game as a positive cost externality.•The enterprises in the market pursue relative profit maximization.•The deeper analyses on stability and bifurcation of Nash equilibrium are studied.•Multi-layer fractal structures, like Arnold's tongues, and their law of arrangement are investigated.•Multi-stability, e.g. coexistence between Nash equilibrium and periodic attractors.
A dynamic Cournot duopoly model with isoelastic demand function and knowledge spillover effects is established, in which the goals of two enterprises are relative profit maximization. Since these two enterprises only have incomplete information of the market, then it is assumed that the enterprises are boundedly rational, and they need to adjust their production decision in next period according to the gradient adjustment mechanism. Sufficient conditions for the existence and stable properties of unique Nash equilibrium can be judged with the help of the Jury criterion. The size of stability region is related to the values of parameters and the system has two different paths to chaos, one is via flip bifurcation and another is via Neimark-Sacker bifurcation. Finally, the complex dynamic characteristics also can be further studied by using numerical simulation, including the structures of multi-layer Arnold's tongues and coexistence of attractors.