IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v143y2021ics0960077920309449.html
   My bibliography  Save this article

Dynamics of a two-dimensional map on nested circles and rings

Author

Listed:
  • Gardini, Laura
  • Sushko, Iryna
  • Tramontana, Fabio

Abstract

We consider a discrete dynamical system, a two-dimensional real map which represents a one-dimensional complex map. Depending on the parameters, its bounded dynamics can be restricted to an invariant circle, cyclic invariant circles, invariant annular regions or disks. We show that on such invariant sets the trajectories are always either periodic of the same period, or quasiperiodic and dense. Moreover, the invariant sets may be transversely attracting or repelling, and undergo the typical cascade of period doubling bifurcations. Homoclinic bifurcations can also occur, leading to chaotic rings, annular regions filled with dense repelling cyclical circles and aperiodic trajectories.

Suggested Citation

  • Gardini, Laura & Sushko, Iryna & Tramontana, Fabio, 2021. "Dynamics of a two-dimensional map on nested circles and rings," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    • Handle: RePEc:eee:chsofr:v:143:y:2021:i:c:s0960077920309449
    DOI: 10.1016/j.chaos.2020.110553
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077920309449
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2020.110553?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Susan Himmelweit, 1976. "A Behavioural Model of Learning in Production," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 43(2), pages 329-346.
    2. Bohm, Volker & Kaas, Leo, 2000. "Differential savings, factor shares, and endogenous growth cycles," Journal of Economic Dynamics and Control, Elsevier, vol. 24(5-7), pages 965-980, June.
    3. Tramontana, Fabio & Sushko, Iryna & Avrutin, Viktor, 2015. "Period adding structure in a 2D discontinuous model of economic growth," Applied Mathematics and Computation, Elsevier, vol. 253(C), pages 262-273.
    4. Matsuyama, Kiminori & Sushko, Iryna & Gardini, Laura, 2016. "Revisiting the model of credit cycles with Good and Bad projects," Journal of Economic Theory, Elsevier, vol. 163(C), pages 525-556.
    5. Matsumoto, Akio, 2005. "Density function of piecewise linear transformation," Journal of Economic Behavior & Organization, Elsevier, vol. 56(4), pages 631-653, April.
    6. Day, Richard H. & Shafer, Wayne, 1987. "Ergodic fluctuations in deterministic economic models," Journal of Economic Behavior & Organization, Elsevier, vol. 8(3), pages 339-361, September.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Tramontana, Fabio & Gardini, Laura & Agliari, Anna, 2011. "Endogenous cycles in discontinuous growth models," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(8), pages 1625-1639.
    2. Tramontana, F. & Gardini, L. & Ferri, P., 2010. "The dynamics of the NAIRU model with two switching regimes," Journal of Economic Dynamics and Control, Elsevier, vol. 34(4), pages 681-695, April.
    3. Tramontana, Fabio & Westerhoff, Frank & Gardini, Laura, 2010. "On the complicated price dynamics of a simple one-dimensional discontinuous financial market model with heterogeneous interacting traders," Journal of Economic Behavior & Organization, Elsevier, vol. 74(3), pages 187-205, June.
    4. Huang, Weihong & Zhang, Yang, 2007. "Distributional dynamics of cautious economic adjustment processes," Journal of Economic Behavior & Organization, Elsevier, vol. 62(3), pages 389-407, March.
    5. Takao Asano & Masanori Yokoo, 2017. "Chaotic Dynamics of a Piecewise Linear Model of Credit Cycles with Imperfect Observability," KIER Working Papers 967, Kyoto University, Institute of Economic Research.
    6. Asano, Takao & Yokoo, Masanori, 2019. "Chaotic dynamics of a piecewise linear model of credit cycles," Journal of Mathematical Economics, Elsevier, vol. 80(C), pages 9-21.
    7. Yokoo, Masanori & Ishida, Junichiro, 2008. "Misperception-driven chaos: Theory and policy implications," Journal of Economic Dynamics and Control, Elsevier, vol. 32(6), pages 1732-1753, June.
    8. Kubin, Ingrid & Zörner, Thomas O. & Gardini, Laura & Commendatore, Pasquale, 2019. "A credit cycle model with market sentiments," Structural Change and Economic Dynamics, Elsevier, vol. 50(C), pages 159-174.
    9. Akio Matsumoto, 1998. "Do Government Subsidies Stabilize Or Destabilize Agricultural Markets?," Contemporary Economic Policy, Western Economic Association International, vol. 16(4), pages 452-466, October.
    10. Huang, Weihong, 1995. "Caution implies profit," Journal of Economic Behavior & Organization, Elsevier, vol. 27(2), pages 257-277, July.
    11. John Stachurski, 2009. "Economic Dynamics: Theory and Computation," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262012774, December.
    12. Kubin, Ingrid & Zörner, Thomas O., 2021. "Credit cycles, human capital and the distribution of income," Journal of Economic Behavior & Organization, Elsevier, vol. 183(C), pages 954-975.
    13. William Barnett & Evgeniya Duzhak, 2010. "Empirical assessment of bifurcation regions within New Keynesian models," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 45(1), pages 99-128, October.
    14. Fernando Suárez, 2021. "Dinámica discreta en un modelo de crecimiento de Solow con ahorro diferencial constante y crecimiento poblacional no constante," Ensayos de Política Económica, Departamento de Investigación Francisco Valsecchi, Facultad de Ciencias Económicas, Pontificia Universidad Católica Argentina., vol. 3(3), pages 25-39, Octubre.
    15. Hashimoto, Ken-ichi & Im, Ryonghun & Kunieda, Takuma & Shibata, Akihisa, 2022. "Financial destabilization," Journal of Mathematical Economics, Elsevier, vol. 103(C).
    16. Alexeeva, Tatyana A. & Kuznetsov, Nikolay V. & Mokaev, Timur N., 2021. "Study of irregular dynamics in an economic model: attractor localization and Lyapunov exponents," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    17. Metcalf, Christopher J., 2008. "The dynamics of the Stiglitz policy in the RSS model," Chaos, Solitons & Fractals, Elsevier, vol. 37(3), pages 652-661.
    18. Paul Beaudry & Dana Galizia & Franck Portier, 2015. "Reviving the Limit Cycle View of Macroeconomic Fluctuations," NBER Working Papers 21241, National Bureau of Economic Research, Inc.
    19. Serena Brianzoni, & Cristiana Mammana, & Elisabetta Michetti,, 2006. "Global attractor in Solow growth model with differential savings and endogenic labor force growth," Working Papers 35-2006, Macerata University, Department of Finance and Economic Sciences, revised Oct 2008.
    20. Ted Theodosopoulos & Muffasir Badshah, 2004. "Short-term equity dynamics and endogenous market fluctuations," Papers math/0406067, arXiv.org, revised Sep 2004.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:143:y:2021:i:c:s0960077920309449. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.