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Global bifurcations in a piecewise-smooth Cournot duopoly game

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  • Tramontana, Fabio
  • Gardini, Laura
  • Puu, Tönu

Abstract

The object of the work is to perform the global analysis of the Cournot duopoly model with isoelastic demand function and unit costs, presented in Puu [2]. The bifurcation of the unique Cournot fixed point is established, which is a resonant case of the Neimark–Sacker bifurcation. New properties associated with the introduction of horizontal branches are evidenced. These properties differ significantly when the constant value is zero or positive and small. The good behavior of the case with positive constant is proved, leading always to positive trajectories. Also when the Cournot fixed point is unstable, stable cycles of any period may exist.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:43:y:2010:i:1:p:15-24
    DOI: 10.1016/j.chaos.2010.07.001
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    1. Puu, T., 1998. "The chaotic duopolists revisited," Journal of Economic Behavior & Organization, Elsevier, vol. 33(3-4), pages 385-394, January.
    2. Agliari, Anna, 2006. "Homoclinic connections and subcritical Neimark bifurcation in a duopoly model with adaptively adjusted productions," Chaos, Solitons & Fractals, Elsevier, vol. 29(3), pages 739-755.
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    8. Angelini, Natascia & Dieci, Roberto & Nardini, Franco, 2009. "Bifurcation analysis of a dynamic duopoly model with heterogeneous costs and behavioural rules," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(10), pages 3179-3196.
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    Cited by:

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    2. Jose S. Cánovas & Anastasiia Panchuk & Tonu Puu, 2015. "Role of reinvestment in a competitive market," Gecomplexity Discussion Paper Series 12, Action IS1104 "The EU in the new complex geography of economic systems: models, tools and policy evaluation", revised Apr 2015.
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    5. Lampart, Marek, 2012. "Stability of the Cournot equilibrium for a Cournot oligopoly model with n competitors," Chaos, Solitons & Fractals, Elsevier, vol. 45(9), pages 1081-1085.
    6. Davide Radi & Laura Gardini, 2023. "Ambiguity aversion as a route to randomness in a duopoly game," Papers 2311.11366, arXiv.org.
    7. Cánovas, Jose S. & Panchuk, Anastasiia & Puu, Tönu, 2015. "Asymptotic dynamics of a piecewise smooth map modelling a competitive market," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 117(C), pages 20-38.
    8. Caravaggio, Andrea & Gori, Luca & Sodini, Mauro, 2022. "Endogenous preferences in a dynamic Cournot duopoly," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    9. Anna Agliari & Pasquale Commendatore & Ilaria Foroni & Ingrid Kubin, 2011. "Border Collision Bifurcations in a Footloose Capital Model with First Nature Firms," Computational Economics, Springer;Society for Computational Economics, vol. 38(3), pages 349-366, October.
    10. Puu, Tonu & Tramontana, Fabio, 2019. "Can Bertrand and Cournot oligopolies be combined?," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 97-107.
    11. Gori, Luca & Sodini, Mauro, 2017. "Price competition in a nonlinear differentiated duopoly," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 557-567.
    12. Agliari, Anna & Pecora, Nicolò & Szuz, Alina, 2022. "Appearance of closed invariant curves in a piecewise Cournot model with advertising," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
    13. Mignot, Sarah & Tramontana, Fabio & Westerhoff, Frank H., 2023. "Complex dynamics in a nonlinear duopoly model with heuristic expectation formation and learning behavior," BERG Working Paper Series 187, Bamberg University, Bamberg Economic Research Group.
    14. 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).
    15. Tramontana, Fabio & Gardini, Laura & Puu, Tönu, 2011. "Mathematical properties of a discontinuous Cournot–Stackelberg model," Chaos, Solitons & Fractals, Elsevier, vol. 44(1), pages 58-70.
    16. Cerboni Baiardi, Lorenzo & Lamantia, Fabio & Radi, Davide, 2015. "Evolutionary competition between boundedly rational behavioral rules in oligopoly games," Chaos, Solitons & Fractals, Elsevier, vol. 79(C), pages 204-225.
    17. Andrea Caravaggio & Mauro Sodini, 2022. "Environmental sustainability, nonlinear dynamics and chaos reloaded: 0 matters!," Discussion Papers 2022/287, Dipartimento di Economia e Management (DEM), University of Pisa, Pisa, Italy.
    18. Fanti, Luciano & Gori, Luca & Sodini, Mauro, 2015. "Nonlinear dynamics in a Cournot duopoly with isoelastic demand," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 108(C), pages 129-143.
    19. 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.
    20. 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).
    21. Fabio Lamantia, 2011. "A Nonlinear Duopoly with Efficient Production-Capacity Levels," Computational Economics, Springer;Society for Computational Economics, vol. 38(3), pages 295-309, October.

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