Research Article Exploration of Complex Dynamics for Cournot Oligopoly Game with Differentiated Products S. S. Askar , 1,2 Mona F. EL-Wakeel , 1,3 and M. A. Alrodaini 1,4 1 Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 354516, Egypt 3 Higher Institute of Computers, Information and Management Technology, Tanta, Egypt 4 Department of Mathematics, College of Science, Al Jouf University, Al Jawf, Saudi Arabia Correspondence should be addressed to S. S. Askar; s.e.a.askar@hotmail.co.uk Received 5 June 2017; Revised 26 November 2017; Accepted 28 December 2017; Published 13 February 2018 Academic Editor: Jos´ e ´ Angel Acosta Copyright © 2018 S. S. Askar et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Tis paper proposes a Cournot game organized by three competing frms adopting bounded rationality. According to the marginal proft in the past time step, each frm tries to update its production using local knowledge. In this game, a frm’s preference is represented by a utility function that is derived from a constant elasticity of substitution (CES) production function. Te game is modeled by a 3-dimensional discrete dynamical system. Te equilibria of the system are numerically studied to detect their complex characteristics due to difculty to get an explicit form for those equilibria. For the proposed utility function, some cases with diferent value parameters are considered. Numerical simulations are used to provide an experimental evidence for the complex behavior of the evolution of the system. Te obtained results show that the system loses its stability due to diferent types of bifurcations. 1. Introduction Te oligopoly market is an efcient market that is dominated by a number of frms which ofer and sell homogeneous or similar products. Such market includes two popular types of models, namely, Cournot (quantity decision) and Bertrand (price decision) models. In Cournot models, the way that frms control their production levels has a critical efect on market outputs. Conversely, in Bertrand models, frms select prices to be their strategic variables to optimize their profts [1, 2]. Practically, the dynamic of an oligopoly game is complex, since each oligopolistic frm must consider both the consumers’ behaviors and the reactions of all competitors [3]. A Cournot triopoly game is an oligopoly market with three players that are in confict and there is no cooperation among them. Te frst idea of a Cournot oligopoly market came to light in 1838 by Cournot [4] who proposed the frst formal oligopoly theory and treated such idea with naive expectations on which each player assumes the last taken values by competitors with no consideration of their prospected reactions in the future [3]. In general, players in an oligopoly market try to enhance their expected proft which in turn is based on matching among the marginal revenue and marginal cost. Furthermore, each player can adjust his output based on selecting his expectation rule from among various available ones such as local bounded rational, adaptive, and naive expectations [3, 5]. Puu [6] conducted some of the frst investigations in such economic games and thus he has concluded that various complex dynamics could result from a Cournot duopoly such as the presence of attractors with a fractal dimension. Various eforts were exerted then to study the dynamics of oligopoly models with considering more frms and various amend- ments [7, 8]. Elsadany and Awad [9] and Askar [10] pre- sented a duopoly Cournot game model with considering the bounded rationality and linear cost and demand functions. Another duopoly game model was studied by Bischi et al. [11], where frms of naive expectations decided their outputs based Hindawi Complexity Volume 2018, Article ID 6526794, 13 pages https://doi.org/10.1155/2018/6526794