IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v139y2020ics0960077920303994.html
   My bibliography  Save this article

On complex dynamic investigations of a piecewise smooth nonlinear duopoly game

Author

Listed:
  • Askar, S.S.
  • Al-khedhairi, A.

Abstract

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.

Suggested Citation

  • Askar, S.S. & Al-khedhairi, A., 2020. "On complex dynamic investigations of a piecewise smooth nonlinear duopoly game," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
  • Handle: RePEc:eee:chsofr:v:139:y:2020:i:c:s0960077920303994
    DOI: 10.1016/j.chaos.2020.110001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077920303994
    Download Restriction: Full text for ScienceDirect subscribers only

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Tramontana, Fabio & Gardini, Laura & Puu, Tönu, 2010. "Global bifurcations in a piecewise-smooth Cournot duopoly game," Chaos, Solitons & Fractals, Elsevier, vol. 43(1), pages 15-24.
    2. Pasquale Commendatore & Ingrid Kubin & Pascal Mossay & Iryna Sushko, 2017. "The role of centrality and market size in a four-region asymmetric new economic geography model," Journal of Evolutionary Economics, Springer, vol. 27(5), pages 1095-1131, November.
    3. Puu, Tonu, 2017. "A New Approach to Modeling Bertrand Duopoly," Review of Behavioral Economics, now publishers, vol. 4(1), pages 51-67, April.
    4. Agiza, H.N. & Elsadany, A.A., 2003. "Nonlinear dynamics in the Cournot duopoly game with heterogeneous players," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 320(C), pages 512-524.
    5. Radi, Davide & Gardini, Laura & Avrutin, Viktor, 2014. "The role of constraints in a segregation model: The symmetric case," Chaos, Solitons & Fractals, Elsevier, vol. 66(C), pages 103-119.
    6. Andaluz, J. & Elsadany, A.A. & Jarne, G., 2017. "Nonlinear Cournot and Bertrand-type dynamic triopoly with differentiated products and heterogeneous expectations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 132(C), pages 86-99.
    7. Hommes,Cars, 2015. "Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems," Cambridge Books, Cambridge University Press, number 9781107564978, January.
    8. Commendatore, Pasquale & Kubin, Ingrid & Petraglia, Carmelo & Sushko, Iryna, 2014. "Regional integration, international liberalisation and the dynamics of industrial agglomeration," Journal of Economic Dynamics and Control, Elsevier, vol. 48(C), pages 265-287.
    9. Matjaž Perc & Zhen Wang, 2010. "Heterogeneous Aspirations Promote Cooperation in the Prisoner's Dilemma Game," PLOS ONE, Public Library of Science, vol. 5(12), pages 1-8, December.
    10. Askar, S.S. & Elettreby, M.F., 2017. "The impact of cost uncertainty on Cournot oligopoly games," Applied Mathematics and Computation, Elsevier, vol. 312(C), pages 169-176.
    11. Peng, Yu & Lu, Qian & Xiao, Yue & Wu, Xue, 2019. "Complex dynamics analysis for a remanufacturing duopoly model with nonlinear cost," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 514(C), pages 658-670.
    12. Agliari, A. & Naimzada, A.K. & Pecora, N., 2016. "Nonlinear dynamics of a Cournot duopoly game with differentiated products," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 1-15.
    13. Shi, Lian & Sheng, Zhaohan & Xu, Feng, 2015. "The dynamics of competition in remanufacturing: A stability analysis," Economic Modelling, Elsevier, vol. 50(C), pages 245-253.
    14. S. S. Askar & A. Al-khedhairi, 2017. "Analysis of Nonlinear Duopoly Games with Product Differentiation: Stability, Global Dynamics, and Control," Discrete Dynamics in Nature and Society, Hindawi, vol. 2017, pages 1-13, June.
    15. Askar, S.S. & Alshamrani, Ahmad M. & Alnowibet, K., 2016. "The arising of cooperation in Cournot duopoly games," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 535-542.
    16. Commendatore, Pasquale & Kubin, Ingrid & Mossay, Pascal & Sushko, Iryna, 2015. "Dynamic agglomeration patterns in a two-country new economic geography model with four regions," Chaos, Solitons & Fractals, Elsevier, vol. 79(C), pages 2-17.
    17. S. S. Askar & Mona F. EL-Wakeel & M. A. Alrodaini, 2018. "Exploration of Complex Dynamics for Cournot Oligopoly Game with Differentiated Products," Complexity, Hindawi, vol. 2018, pages 1-13, February.
    18. Zhu, Peican & Wang, Xinyu & Li, Shudong & Guo, Yangming & Wang, Zhen, 2019. "Investigation of epidemic spreading process on multiplex networks by incorporating fatal properties," Applied Mathematics and Computation, Elsevier, vol. 359(C), pages 512-524.
    19. Askar, S.S., 2018. "Tripoly Stackelberg game model: One leader versus two followers," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 301-311.
    20. Commendatore, Pasquale & Kubin, Ingrid & Sushko, Iryna, 2015. "Typical bifurcation scenario in a three region identical New Economic Geography model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 108(C), pages 63-80.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Askar, S.S. & Al-khedhairi, A., 2020. "The dynamics of a business game: A 2D-piecewise smooth nonlinear map," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).
    2. Peng, Yu & Xiao, Yue & Lu, Qian & Wu, Xue & Zhao, Yueru, 2020. "Chaotic dynamics in Cournot duopoly model with bounded rationality based on relative profit delegation maximization," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 560(C).
    3. Peng, Yu & Lu, Qian & Wu, Xue & Zhao, Yueru & Xiao, Yue, 2020. "Dynamics of Hotelling triopoly model with bounded rationality," Applied Mathematics and Computation, Elsevier, vol. 373(C).
    4. S. S. Askar & A. Al-khedhairi, 2019. "Analysis of a Four-Firm Competition Based on a Generalized Bounded Rationality and Different Mechanisms," Complexity, Hindawi, vol. 2019, pages 1-12, May.
    5. Schmitt, Noemi & Tuinstra, Jan & Westerhoff, Frank, 2017. "Side effects of nonlinear profit taxes in an evolutionary market entry model: Abrupt changes, coexisting attractors and hysteresis problems," Journal of Economic Behavior & Organization, Elsevier, vol. 135(C), pages 15-38.
    6. Xin, Baogui & Peng, Wei & Kwon, Yekyung, 2020. "A discrete fractional-order Cournot duopoly game," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 558(C).
    7. Mikhail Anufriev & Davide Radi & Fabio Tramontana, 2018. "Some reflections on past and future of nonlinear dynamics in economics and finance," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 41(2), pages 91-118, November.
    8. Garmani, Hamid & Ait Omar, Driss & El Amrani, Mohamed & Baslam, Mohamed & Jourhmane, Mostafa, 2020. "Analysis of a dynamics duopoly game with two content providers," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    9. Noemi Schmitt & Jan Tuinstra & Frank Westerhoff, 2018. "Stability and welfare effects of profit taxes within an evolutionary market interaction model," Review of International Economics, Wiley Blackwell, vol. 26(3), pages 691-708, August.
    10. Yu Yu & Weisheng Yu, 2019. "The Complexion of Multi-period Stackelberg Triopoly Game with Bounded Rationality," Computational Economics, Springer;Society for Computational Economics, vol. 53(1), pages 457-478, January.
    11. Yang, Xuenan & Peng, Yu & Xiao, Yue & Wu, Xue, 2019. "Nonlinear dynamics of a duopoly Stackelberg game with marginal costs," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 185-191.
    12. Askar, S.S., 2018. "Tripoly Stackelberg game model: One leader versus two followers," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 301-311.
    13. Gian Italo Bischi & Fabio Lamantia & Davide Radi, 2018. "Evolutionary oligopoly games with heterogeneous adaptive players," Chapters, in: Luis C. Corchón & Marco A. Marini (ed.), Handbook of Game Theory and Industrial Organization, Volume I, chapter 12, pages 343-370, Edward Elgar Publishing.
    14. Peng, Yu & Lu, Qian & Xiao, Yue & Wu, Xue, 2019. "Complex dynamics analysis for a remanufacturing duopoly model with nonlinear cost," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 514(C), pages 658-670.
    15. Cerboni Baiardi, Lorenzo & Naimzada, Ahmad K., 2019. "An oligopoly model with rational and imitation rules," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 156(C), pages 254-278.
    16. Pasquale Commendatore & Ingrid Kubin & Pascal Mossay & Iryna Sushko, 2015. "Dynamic agglomeration patterns in a 2-country NEG model with 4 regions," Gecomplexity Discussion Paper Series 10, Action IS1104 "The EU in the new complex geography of economic systems: models, tools and policy evaluation", revised Feb 2015.
    17. Lorenzo Cerboni Baiardi & Ahmad K. Naimzada, 2018. "An evolutionary model with best response and imitative rules," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 41(2), pages 313-333, November.
    18. Lorenzo Cerboni Baiardi & Ahmad K. Naimzada, 2019. "An evolutionary Cournot oligopoly model with imitators and perfect foresight best responders," Metroeconomica, Wiley Blackwell, vol. 70(3), pages 458-475, July.
    19. Ahmed, E. & Elsadany, A.A. & Puu, Tonu, 2015. "On Bertrand duopoly game with differentiated goods," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 169-179.
    20. Cerboni Baiardi, Lorenzo & Naimzada, Ahmad K., 2018. "An oligopoly model with best response and imitation rules," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 193-205.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:139:y:2020:i:c:s0960077920303994. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Thayer, Thomas R.). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.