Stability and dynamic behaviors of a limited monopoly with a gradient adjustment mechanism

In this paper, a limited monopoly game proposed by Puu with a gradient adjustment mechanism is reconsidered using bifurcation theory and several tools based on symbolic computations. To the best of our knowledge, the complete stability conditions of the equilibria of this game are obtained for the first time. We explore the existence and stability of periodic orbits and demonstrate the occurrence of an infinite number of four-cycles, which corrects the erroneous results derived by Al-Hdaibat and others. We prove that the model possesses no two-snapback repellers or three-snapback repellers, the existence of which is a sufficient condition for the emergence of chaos. Furthermore, fold, Neimark-Sacker, cusp, and 1:4 resonance bifurcations are analyzed in detail and the corresponding dynamics are presented. Symmetric and complex dynamic phenomena are observed through our numerical simulations.