Chaos and periodic solutions in a dynamic monopoly model

T. Puu [2,3] proposed a 2D monopoly model with cubic price and quadratic marginal cost functions. He provides incomplete information on the existence of cycles of period 4 and the chaotic behavior in his model [3]. Though the recent literature still deals with simplified versions of the monopoly model of T. Puu, and none of them analyzes the dynamic behavior of the T. Puu model in detail. In a recent paper [1], we reconsider the dynamic monopoly model. We present fundamental corrections to the fixed point stability analysis presented in [3]. By simulations, the existence of solutions of period 4, 5, 10, 13, 17 and the chaotic behavior are investigated. Continuation and bifurcation analysis is used to get information about the stability of 5,10,13,17-cycles under parameter variation. In all regions, further period-doubling bifurcations are found which implies the existence of orbits with higher periods as well. A general formula for solutions of period 4 is derived. Among other things, we discuss the symmetry properties of these solutions. The analytical stability analysis for the 4-cycles proves that they are never linearly asymptotically stable. Therefore, the stability of the 4-cycle is investigated by studying the effect of small displacements applied to the eigenvector corresponding to the eigenvalue located at the stability boundary. This work, combined with simulation and the basin of attraction analysis for the 4-cycle allows us to determine the stability region of the 4-cycle. This region is larger than the one obtained in [3] which is based on an incorrect linear stability analysis. We analyze the chaotic behavior of the monopoly model by computing the largest Lyapunov exponents. This analysis confirms our results. References [1] Bashir Al-Hdaibat, Willy Govaerts, , Niels Neirynck. On periodic and chaotic behavior in a two-dimensional monopoly model, Chaos, Solitons & Fractals, 70 (2015), pp. 27–37. [2] T. Puu, The chaotic monopolist, Chaos, Solitons & Fractals, 5 (1) (1995), pp. 35–44. [3] T. Puu, Attractors, bifurcations, and chaos: nonlinear phenomena in economics, Springer-Verlag, Berlin (2000).