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On periodic and chaotic behavior in a two-dimensional monopoly model

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  • Al-Hdaibat, Bashir
  • Govaerts, Willy
  • Neirynck, Niels

Abstract

We study the monopoly model of T. Puu with cubic price and quadratic marginal cost functions. A numerical continuation method is used to compute branches of solutions of period 5, 10, 13 and 17 and to determine the stability regions of these solutions. General formulas for solutions of period 4 are derived analytically. We show that the solutions of period 4 are never linearly asymptotically stable. A nonlinear stability criterion is combined with basin of attraction analysis and simulation to determine the stability region of the 4-cycles. This corrects the erroneous linear stability analysis in previous studies of the model. The chaotic and periodic behavior of the monopoly model is further analyzed by computing the largest Lyapunov exponents, and this confirms the above mentioned results.

Suggested Citation

  • Al-Hdaibat, Bashir & Govaerts, Willy & Neirynck, Niels, 2015. "On periodic and chaotic behavior in a two-dimensional monopoly model," Chaos, Solitons & Fractals, Elsevier, vol. 70(C), pages 27-37.
    • Handle: RePEc:eee:chsofr:v:70:y:2015:i:c:p:27-37
    DOI: 10.1016/j.chaos.2014.10.010
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    References listed on IDEAS

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    1. Askar, S.S., 2013. "On complex dynamics of monopoly market," Economic Modelling, Elsevier, vol. 31(C), pages 586-589.
    2. Grandmont, Jean-Michel, 2008. "Nonlinear difference equations, bifurcations and chaos: An introduction," Research in Economics, Elsevier, vol. 62(3), pages 122-177, September.
    3. Ahmed, E. & Elettreby, M.F. & Hegazi, A.S., 2006. "On Puu’s incomplete information formulation for the standard and multi-team Bertrand game," Chaos, Solitons & Fractals, Elsevier, vol. 30(5), pages 1180-1184.
    4. Matsumoto, Akio & Szidarovszky, Ferenc, 2014. "Discrete and continuous dynamics in nonlinear monopolies," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 632-642.
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    Cited by:

    1. Xiaoliang Li, 2021. "Analysis of stability and bifurcation for two heterogeneous triopoly games with the isoelastic demand," Papers 2112.05950, arXiv.org.
    2. Xiaoliang Li & Jiacheng Fu & Wei Niu, 2023. "Complex dynamics of knowledgeable monopoly models with gradient mechanisms," Papers 2301.01497, arXiv.org.
    3. Lahmiri, Salim, 2017. "On fractality and chaos in Moroccan family business stock returns and volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 473(C), pages 29-39.
    4. Li, Xiaoliang & Yang, Jing & Zhang, Ally Quan, 2025. "Influence of price elasticity of demand on monopoly games under different returns to scale," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 233(C), pages 75-98.
    5. Lahmiri, Salim, 2017. "Investigating existence of chaos in short and long term dynamics of Moroccan exchange rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 465(C), pages 655-661.
    6. Lahmiri, Salim, 2017. "A study on chaos in crude oil markets before and after 2008 international financial crisis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 466(C), pages 389-395.
    7. Bashir Al-Hdaibat & A. Alameer, 2025. "Bifurcation and Chaos in a Nonlinear Population Model with Delayed Nonlinear Interactions," Mathematics, MDPI, vol. 13(13), pages 1-23, June.
    8. Cavalli, Fausto & Naimzada, Ahmad, 2015. "Effect of price elasticity of demand in monopolies with gradient adjustment," Chaos, Solitons & Fractals, Elsevier, vol. 76(C), pages 47-55.
    9. Li, Xiaoliang & Li, Bo & Liu, Li, 2023. "Stability and dynamic behaviors of a limited monopoly with a gradient adjustment mechanism," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).

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