IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v44y2011i4p248-259.html
   My bibliography  Save this article

Border collision bifurcation curves and their classification in a family of 1D discontinuous maps

Author

Listed:
  • Gardini, Laura
  • Tramontana, Fabio

Abstract

In this paper we consider a one-dimensional piecewise linear discontinuous map in canonical form, which may be used in several physical and engineering applications as well as to model some simple financial markets. We classify three different kinds of possible dynamic behaviors associated with the stable cycles. One regime (i) is the same existing in the continuous case and it is characterized by periodicity regions following the period increment by 1 rule. The second one (ii) is the regime characterized by periodicity regions of period increment higher than 1 (we shall see examples with 2 and 3), and by bistability. The third one (iii) is characterized by infinitely many periodicity regions of stable cycles, which follow the period adding structure, and multistability cannot exist. The analytical equations of the border collision bifurcation curves bounding the regions of existence of stable cycles are determined by using a new approach.

Suggested Citation

  • Gardini, Laura & Tramontana, Fabio, 2011. "Border collision bifurcation curves and their classification in a family of 1D discontinuous maps," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 248-259.
    • Handle: RePEc:eee:chsofr:v:44:y:2011:i:4:p:248-259
    DOI: 10.1016/j.chaos.2011.02.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S096007791100018X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2011.02.001?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. 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.
    2. Day, Richard H, 1982. "Irregular Growth Cycles," American Economic Review, American Economic Association, vol. 72(3), pages 406-414, June.
    3. Viktor Avrutin & Michael Schanz & Björn Schenke, 2011. "Coexistence of the Bandcount-Adding and Bandcount-Increment Scenarios," Discrete Dynamics in Nature and Society, Hindawi, vol. 2011, pages 1-30, March.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Giovanni Campisi & Silvia Muzzioli & Fabio Tramontana, 2021. "Uncertainty about fundamental, pessimistic and overconfident traders: a piecewise-linear maps approach," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 44(2), pages 707-726, December.
    2. Foroni, Ilaria & Avellone, Alessandro & Panchuk, Anastasiia, 2015. "Sudden transition from equilibrium stability to chaotic dynamics in a cautious tâtonnement model," Chaos, Solitons & Fractals, Elsevier, vol. 79(C), pages 105-115.
    3. Matsuo, Akihito & Asahara, Hiroyuki & Kousaka, Takuji, 2012. "Bifurcation structure of chaotic attractor in switched dynamical systems with spike noise," Chaos, Solitons & Fractals, Elsevier, vol. 45(6), pages 795-804.
    4. Cerboni Baiardi, Lorenzo & Naimzada, Ahmad K. & Panchuk, Anastasiia, 2020. "Endogenous desired debt in a Minskyan business model," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    5. Roya Makrooni & Laura Gardini, 2015. "Bifurcation structures in a family of one-dimensional linear-power discontinuous maps," Gecomplexity Discussion Paper Series 7, Action IS1104 "The EU in the new complex geography of economic systems: models, tools and policy evaluation", revised Jan 2015.
    6. Brianzoni, Serena & Campisi, Giovanni, 2020. "Dynamical analysis of a financial market with fundamentalists, chartists, and imitators," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    7. Giovanni Campisi & Silvia Muzzioli & Fabio Tramontana, 2021. "Uncertainty about fundamental and pessimistic traders: a piecewise-linear maps approach," Department of Economics 0186, University of Modena and Reggio E., Faculty of Economics "Marco Biagi".

    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. Gardini, Laura & Tramontana, Fabio, 2012. "Structurally unstable regular dynamics in 1D piecewise smooth maps, and circle maps," Chaos, Solitons & Fractals, Elsevier, vol. 45(11), pages 1328-1342.
    2. Schmitt, Noemi & Tuinstra, Jan & Westerhoff, Frank, 2017. "Side effects of nonlinear profit taxes in an evolutionary market entry model: Abrupt changes, coexisting attractors and hysteresis problems," Journal of Economic Behavior & Organization, Elsevier, vol. 135(C), pages 15-38.
    3. Viktor Avrutin & Iryna Sushko & Fabio Tramontana, 2014. "Bifurcation Structure in a Bimodal Piecewise Linear Business Cycle Model," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-12, November.
    4. Gomes, Orlando, 2008. "Too much of a good thing: Endogenous business cycles generated by bounded technological progress," Economic Modelling, Elsevier, vol. 25(5), pages 933-945, September.
    5. Michele Boldrin, 1988. "Persistent Oscillations and Chaos in Dynamic Economic Models: Notes for a Survey," UCLA Economics Working Papers 458A, UCLA Department of Economics.
    6. Nishimura, Kazuo & Yano, Makoto, 1995. "Durable capital and chaos in competitive business cycles," Journal of Economic Behavior & Organization, Elsevier, vol. 27(2), pages 165-181, July.
    7. 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.
    8. Verónica ACURIO VASCONEZ & David DESMARCHELIER & Romain RESTOUT, 2024. "Pollution, Endogenous Capital Depreciation, and Growth Dynamics," Working Papers of BETA 2024-01, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.
    9. Hallegatte, Stéphane & Ghil, Michael, 2008. "Natural disasters impacting a macroeconomic model with endogenous dynamics," Ecological Economics, Elsevier, vol. 68(1-2), pages 582-592, December.
    10. 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.
    11. Tamotsu Onozaki, 2018. "Nonlinearity, Bounded Rationality, and Heterogeneity," Springer Books, Springer, number 978-4-431-54971-0, September.
    12. Roa Maria J & Vazquez Francisco Jose & Saura Dulce, 2008. "Unemployment and Economic Growth Cycles," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 12(2), pages 1-21, May.
    13. F. Cavalli & A. Naimzada & N. Pecora, 2022. "A stylized macro-model with interacting real, monetary and stock markets," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 17(1), pages 225-257, January.
    14. Weihong Huang & Huanhuan Zheng & Wai-Mun Chia, 2013. "Asymmetric returns, gradual bubbles and sudden crashes," The European Journal of Finance, Taylor & Francis Journals, vol. 19(5), pages 420-437, May.
    15. Schmitt, Noemi & Westerhoff, Frank, 2014. "Speculative behavior and the dynamics of interacting stock markets," Journal of Economic Dynamics and Control, Elsevier, vol. 45(C), pages 262-288.
    16. Hallegatte, Stéphane & Ghil, Michael & Dumas, Patrice & Hourcade, Jean-Charles, 2008. "Business cycles, bifurcations and chaos in a neo-classical model with investment dynamics," Journal of Economic Behavior & Organization, Elsevier, vol. 67(1), pages 57-77, July.
    17. Tomohiro Uchiyama, 2023. "A necessary and sufficient condition for the existence of chaotic dynamics in a neoclassical growth model with a pollution effect," Papers 2311.03594, arXiv.org.
    18. Noemi Schmitt & Frank Westerhoff, 2017. "Heterogeneity, spontaneous coordination and extreme events within large-scale and small-scale agent-based financial market models," Journal of Evolutionary Economics, Springer, vol. 27(5), pages 1041-1070, November.
    19. Dercole, Fabio & Radi, Davide, 2020. "Does the “uptick rule” stabilize the stock market? Insights from adaptive rational equilibrium dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    20. Matsumoto, Akio, 1998. "Non-linear structure of a Metzlerian inventory cycle model," Journal of Economic Behavior & Organization, Elsevier, vol. 33(3-4), pages 481-492, January.

    More about this item

    Statistics

    Access and download statistics

    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:44:y:2011:i:4:p:248-259. 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.