Study on dynamical properties and simulation of a four-dimensional nonlinear discrete dynamics

Abstract

We study a nonlinear discrete dynamic game model of an oligopoly market. In order to study competition process of the players, the paper considers a Bertrand model with bounded rationality. A game with four oligopolies is modeled by a four-dimensional nonlinear difference equation set. The stability of the equilibrium point are discussed. We demonstrate rich dynamical behaviors of the system. The chaotic features are justified numerically via bifurcation diagrams, the maximal Lyapunov exponents and the system's sensitive dependence on initial conditions. It is demonstrated the increasing of price adjustment parameters might change stability of the Nash equilibrium and cause bifurcation and chaos. Different from the former literatures, we find that chaos maybe caused by interaction of some elements of the system. On that basis, the main factors might lead the system to chaos are discussed.