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Complexity Analysis of a 2D‐Piecewise Smooth Duopoly Model: New Products versus Remanufactured Products

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  • S. S. Askar
  • A. Al-khedhairi

Abstract

Recent studies on remanufacturing duopoly games have handled them as smooth maps and have observed that the bifurcation types that occurred in such maps belong to generic classes like period‐doubling or Neimark‐Sacker bifurcations. Since those games yield piecewise smooth maps, their bifurcations belong to the so‐called border‐collision bifurcations, which occur when the map’s fixed points cross the borderline between the smooth regions in the phase space. In the current paper, we present a proper systematic analysis of the local stability of the map’s fixed points both analytically and numerically. This includes studying the border‐collision bifurcation depending on the map’s parameters. We present different multistability scenarios of the dynamics of the game’s map and show different types of periodic cycles and chaotic attractors that jump from one region to another or just cross the borderline in the phase space.

Suggested Citation

  • S. S. Askar & A. Al-khedhairi, 2022. "Complexity Analysis of a 2D‐Piecewise Smooth Duopoly Model: New Products versus Remanufactured Products," Complexity, John Wiley & Sons, vol. 2022(1).
    • Handle: RePEc:wly:complx:v:2022:y:2022:i:1:n:8856009
    DOI: 10.1155/2022/8856009
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    References listed on IDEAS

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    1. Gori, Luca & Sodini, Mauro, 2017. "Price competition in a nonlinear differentiated duopoly," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 557-567.
    2. Anna Agliari & Laura Gardini & Tonu Puu, 2006. "Global Bifurcations In Duopoly When The Cournot Point Is Destabilized Via A Subcritical Neimark Bifurcation," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 8(01), pages 1-20.
    3. Hommes,Cars, 2015. "Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems," Cambridge Books, Cambridge University Press, number 9781107564978, August.
    4. 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.
    5. Geraldo Ferrer & Jayashankar M. Swaminathan, 2006. "Managing New and Remanufactured Products," Management Science, INFORMS, vol. 52(1), pages 15-26, January.
    6. S. S. Askar & A. Al-khedhairi, 2019. "Cournot Duopoly Games: Models and Investigations," Mathematics, MDPI, vol. 7(11), pages 1-15, November.
    7. Elsadany, A.A., 2017. "Dynamics of a Cournot duopoly game with bounded rationality based on relative profit maximization," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 253-263.
    8. Li, Yan & Wang, Lidong, 2019. "Chaos in a duopoly model of technological innovation with bounded rationality based on constant conjectural variation," Chaos, Solitons & Fractals, Elsevier, vol. 120(C), pages 116-126.
    9. 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.
    10. Nirvikar Singh & Xavier Vives, 1984. "Price and Quantity Competition in a Differentiated Duopoly," RAND Journal of Economics, The RAND Corporation, vol. 15(4), pages 546-554, Winter.
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