Bifurcation analysis and complex dynamics of a Kopel triopoly model

Abstract

In this paper, a novel type of generalized Kopel triopoly model is presented to reveal the complex dynamics and transitions between different dynamic behaviors. First, based on microeconomic theory, the construction process of the Kopel triopoly model is explained in detail. Second, the existence and stability of fixed points are derived and the corresponding transition processes are presented clearly for some fixed parameters. Bifurcation sets and the critical normal forms of different types of bifurcations are computed to detect possible dynamics. Finally, numerical simulations are conducted to derive representative orbits, chaotic indicators, Lyapunov exponents, bifurcation continuation, and one- (two-) dimensional parameter spaces for the triopoly model. For example, periodic structures are presented with the numbers of corresponding periods. Some of the derived periodic orbits and Lyapunov exponents are plotted to highlight potentially stable and unstable dynamic behaviors. The results demonstrate the complexity of the Kopel triopoly game and corresponding mechanisms.