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Some global bifurcations related to the appearance of closed invariant curves

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  • Agliari, Anna
  • Gardini, Laura
  • Puu, Tönu

Abstract

In this paper, we consider a two-dimensional map (a duopoly game) in which the fixed point is destabilized via a subcritical Neimark–Hopf (N–H) bifurcation. Our aim is to investigate, via numerical examples, some global bifurcations associated with the appearance of repelling closed invariant curves involved in the Neimark–Hopf bifurcations. We shall see that the mechanism is not unique, and that it may be related to homoclinic connections of a saddle cycle, that is to a closed invariant curve formed by the merging of a branch of the stable set of the saddle with a branch of the unstable set of the same saddle. This will be shown by analyzing the bifurcations arising inside a periodicity tongue, i.e., a region of the parameter space in which an attracting cycle exists.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:matcom:v:68:y:2005:i:3:p:201-219
    DOI: 10.1016/j.matcom.2004.12.003
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    References listed on IDEAS

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    1. Bischi, Gian Italo & Kopel, Michael, 2001. "Equilibrium selection in a nonlinear duopoly game with adaptive expectations," Journal of Economic Behavior & Organization, Elsevier, vol. 46(1), pages 73-100, September.
    2. Rand, David, 1978. "Exotic phenomena in games and duopoly models," Journal of Mathematical Economics, Elsevier, vol. 5(2), pages 173-184, September.
    3. Kind, Christoph, 1999. "Remarks on the economic interpretation of Hopf bifurcations," Economics Letters, Elsevier, vol. 62(2), pages 147-154, February.
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    Citations

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    Cited by:

    1. 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.
    2. Michael Wegener & Frank Westerhoff, 2012. "Evolutionary competition between prediction rules and the emergence of business cycles within Metzler’s inventory model," Journal of Evolutionary Economics, Springer, vol. 22(2), pages 251-273, April.
    3. 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.
    4. Fanti, Luciano & Gori, Luca & Sodini, Mauro, 2012. "Nonlinear dynamics in a Cournot duopoly with relative profit delegation," MPRA Paper 37834, University Library of Munich, Germany.
    5. 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.
    6. Bignami Fernando & Agliari Anna, 2010. "Synchronization and On-Off Intermittency Phenomena in a Market Model with Complementary Goods and Adaptive Expectations," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 14(2), pages 1-31, March.
    7. Yu, Weisheng & Yu, Yu, 2014. "The complexion of dynamic duopoly game with horizontal differentiated products," Economic Modelling, Elsevier, vol. 41(C), pages 289-297.
    8. Fanti, Luciano & Gori, Luca, 2012. "The dynamics of a differentiated duopoly with quantity competition," Economic Modelling, Elsevier, vol. 29(2), pages 421-427.
    9. repec:eee:apmaco:v:281:y:2016:i:c:p:1-15 is not listed on IDEAS
    10. Agliari, Anna & Dieci, Roberto & Gardini, Laura, 2007. "Homoclinic tangles in a Kaldor-like business cycle model," Journal of Economic Behavior & Organization, Elsevier, vol. 62(3), pages 324-347, March.
    11. Agliari, Anna & Rillosi, Francesco & Vachadze, George, 2015. "Credit market imperfection, financial market globalization, and catastrophic transition," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 108(C), pages 41-62.
    12. Li, Jiaorui & Ren, Zhengzheng & Wang, Zuoren, 2008. "Response of nonlinear random business cycle model with time delay state feedback," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(23), pages 5844-5851.
    13. Tramontana, Fabio, 2010. "Heterogeneous duopoly with isoelastic demand function," Economic Modelling, Elsevier, vol. 27(1), pages 350-357, January.
    14. 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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