Complexity and chaos control in a Cournot duopoly model based on bounded rationality and relative profit maximization

The bifurcation analysis of a discrete two-dimensional map that models a Cournot duopoly model, based on bounded rationality and relative profit maximization, is studied. Rigorous proofs of flip bifurcation, Neimark–Sacker bifurcation, 1:2 resonance and Li–Yorke chaos in the Marotto sense are presented in detail. In addition, numerical simulations such as bifurcation transition diagrams, plots of the maximum Lyapunov exponent, invariant curves, time series plots and chaotic attractors, are performed to support theoretical results. We provide parameter values that generate various types of bifurcation and chaotic phenomena rather than just simple numerical examples. The obtained results show game participants can play output dynamics for an extended period and easily fall into chaos if the output adjustment rate of one or both sides is excessive. Finally, two control methods are successfully proposed to control chaos from two perspectives of the players.