Controlling chaos and mixed mode oscillations in a Bertrand duopoly game with homogeneous expectations and quadratic cost functions

Abstract

This paper explores the dynamic behavior of a Bertrand duopoly game involving boundedly rational firms and a quadratic cost function. The study delves into the nonlinear and complex dynamics that appear when the Bertrand–Nash equilibrium point loses its stability as both the speed of adjustment and the differentiation measure between the products increase, characterized by a period-doubling bifurcation. Subsequently, the system exhibits chaos and mixed-mode oscillations with unpredictable patterns through a sequence of flip bifurcations, as demonstrated by numerical analyses. The application of state feedback control successfully stabilizes the system at the Bertrand–Nash equilibrium point. This control method defines three stability boundaries, outlining a triangular region in parameters space. Each line corresponds to specific scenarios influencing overall stability, with intersections indicating the stability region.