On complex dynamic investigations of a piecewise smooth nonlinear duopoly game

• Published in: Chaos, solitons and fractals Vol. 139; p. 110001
• Main Authors: Askar, S.S., Al-khedhairi, A.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.10.2020
• Subjects: Chaotic attractors; Piecewise system; Remanufacturing; Piecewise system; Remanufacturing; Chaotic attractors
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2020.110001

•We introduce a remanufacturing duopoly game based on a piecewise smooth map.•The competed firms in this game adopt the bounded rationality mechanism.•The second firm remanufactures the returned products and then sells them again in the market.•The equilibrium points of the game lose their stability through Flip and Neimark-Sacker bifurcations.•The numerical simulation conducted shows periodicity and multi-stability of the system’s game. In this paper, we introduce a nonlinear duopoly game which is modelled by a piecewise smooth map. The inverse demand functions of the game are derived from a proposed nonlinear utility function. In the game, we consider that the second firm remanufactures returned products and then sell them again in the market. This means that the second firm receives returned products after the first firm presents its products in the market. A discrete-time piecewise smooth map is used to describe this game based on bounded rationality mechanism. The phase space of the map is divided into two regions separated by a border curve. The map’s fixed points in both regions are obtained. Their conditions of stability are discussed. We carry out numerical experiments to confirm the local stability of these fixed points and identify types of bifurcations by which they become unstable. Through simulation we highlight on periodic cycles and chaotic attractors that jump from region to other or cross the border curve in the phase space.