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Homoclinic connections and subcritical Neimark bifurcation in a duopoly model with adaptively adjusted productions

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  • Agliari, Anna

Abstract

In this paper we study some global bifurcations arising in the Puu’s oligopoly model when we assume that the producers do not adjust to the best reply but use an adaptive process to obtain at each step the new production. Such bifurcations cause the appearance of a pair of closed invariant curves, one attracting and one repelling, this latter being involved in the subcritical Neimark bifurcation of the Cournot equilibrium point. The aim of the paper is to highlight the relationship between the global bifurcations causing the appearance/disappearance of two invariant closed curves and the homoclinic connections of some saddle cycle, already conjectured in [Agliari A, Gardini L, Puu T. Some global bifurcations related to the appearance of closed invariant curves. Comput Math Simul 2005;68:201–19]. We refine the results obtained in such a paper, showing that the appearance/disappearance of closed invariant curves is not necessarily related to the existence of an attracting cycle. The characterization of the periodicity tongues (i.e. a region of the parameter space in which an attracting cycle exists) associated with a subcritical Neimark bifurcation is also discussed.

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  • 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.
    • Handle: RePEc:eee:chsofr:v:29:y:2006:i:3:p:739-755
    DOI: 10.1016/j.chaos.2005.08.105
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    1. Kind, Christoph, 1999. "Remarks on the economic interpretation of Hopf bifurcations," Economics Letters, Elsevier, vol. 62(2), pages 147-154, February.
    2. Agliari, Anna & Gardini, Laura & Puu, Tönu, 2005. "Some global bifurcations related to the appearance of closed invariant curves," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 68(3), pages 201-219.
    3. Rand, David, 1978. "Exotic phenomena in games and duopoly models," Journal of Mathematical Economics, Elsevier, vol. 5(2), pages 173-184, September.
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    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. Ding, Zhanwen & Shi, Guiping, 2009. "Cooperation in a dynamical adjustment of duopoly game with incomplete information," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 989-993.
    3. Ilaria Foroni & Anna Agliari, 2008. "Complex Price Dynamics in a Financial Market with Imitation," Computational Economics, Springer;Society for Computational Economics, vol. 32(1), pages 21-36, September.
    4. Bischi, Gian Italo & Cerboni Baiardi, Lorenzo, 2015. "A dynamic marketing model with best reply and inertia," Chaos, Solitons & Fractals, Elsevier, vol. 79(C), pages 145-156.
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    7. Alina Szuz & Anna Agliari & Tonu Puu, 2015. "A Cournot triopoly with two identical producers," Gecomplexity Discussion Paper Series 13, Action IS1104 "The EU in the new complex geography of economic systems: models, tools and policy evaluation", revised Aug 2015.
    8. Lines, Marji & Westerhoff, Frank, 2010. "Inflation expectations and macroeconomic dynamics: The case of rational versus extrapolative expectations," Journal of Economic Dynamics and Control, Elsevier, vol. 34(2), pages 246-257, February.
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    10. Puu, Tonu & Tramontana, Fabio, 2019. "Can Bertrand and Cournot oligopolies be combined?," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 97-107.
    11. Foroni, Ilaria & Agliari, Anna, 2011. "Complex dynamics associated with the appearance/disappearance of invariant closed curves," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(8), pages 1640-1655.
    12. Panchuk, A. & Puu, T., 2015. "Oligopoly model with recurrent renewal of capital revisited," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 108(C), pages 119-128.
    13. 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.
    14. 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).
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