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Front. Phys. | doi: 10.3389/fphy.2019.00084

A continuous time Bertrand duopoly game with fractional delay and conformable derivative: Modelling, discretization process, Hopf bifurcation and chaos

 Baogui Xin1*, Wei Peng1 and  Luca Guerrini2
  • 1Shandong University of Science and Technology, China
  • 2Marche Polytechnic University, Italy

The purpose of this paper is threefold. First, we present a discretization process to obtain numerical solutions of a conformable fractional-order system with delays. Second, we extend the classical Bertrand duopoly game with integer delays to that with fractional delays. Third, we extend the game based on ordinary differential derivative to that based on conformable fractional-order derivative. Finally, we analyze the local stability, Hopf bifurcation, and chaos of the proposed game model.

Keywords: Conformable calculus, Fractional-order Bertrand game, Fractional delay, 0-1 test for chaos, Hopf bifurcation

Received: 11 Apr 2019; Accepted: 16 May 2019.

Edited by:

Carla M. Pinto, Instituto Superior de Engenharia do Porto (ISEP), Portugal

Reviewed by:

Devendra Kumar, University of Rajasthan, India
Francisco Gomez, Centro Nacional de Investigación y Desarrollo Tecnológico, Mexico
Amin Jajarmi, University of Bojnord, Iran
Zhen Wang, Shandong University of Science and Technology, China
Mostafa Eslami, University of Mazandaran, Iran  

Copyright: © 2019 Xin, Peng and Guerrini. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Prof. Baogui Xin, Shandong University of Science and Technology, Qingdao, China, xin@tju.edu.cn