IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v199y2025ip1s0960077925006824.html
   My bibliography  Save this article

A new kind of T-point in the Lorenz system with a different bifurcation set

Author

Listed:
  • Algaba, A.
  • Fernández-Sánchez, F.
  • Merino, M.
  • Rodríguez-Luis, A.J.

Abstract

In this work we find a new kind of T-point (or Bykov point) in the Lorenz system. At this codimension-two degeneracy, a heteroclinic cycle connects the origin (when it is a real saddle) and non-trivial equilibria (when they are saddle-focus). We observe that it presents a noteworthy geometric difference from the “classical” T-point, known since the 1980s in the Lorenz system. Because the dominant eigenvalue of the two-dimensional manifold at the origin changes, a variation in the direction of the corresponding heteroclinic orbit occurs near this equilibrium. Simultaneously, there is an important change in the bifurcation set, not previously found in the literature. While at the classical T-point the homoclinic and heteroclinic curves of non-trivial equilibria arise as half-lines in the same direction (as predicted by the well-known model of Glendinning and Sparrow), now these global bifurcation curves emerge in opposite directions. To justify this change we build a theoretical model with suitable Poincaré sections in a tubular environment of the heteroclinic cycle. Finally, by introducing a fourth parameter into the Lorenz system (a new quadratic term in its third equation), we show how the classical T-point can also lead to the new bifurcation set. This transition through a nongeneric situation (which occurs when the Jacobian matrix at the origin has a double eigenvalue) implies the existence of a codimension-three degenerate T-point. We find this bifurcation in the Lorenz-like system considered and illustrate how the bifurcation sets evolve by analyzing parallel parameter planes on both sides of the degeneracy.

Suggested Citation

  • Algaba, A. & Fernández-Sánchez, F. & Merino, M. & Rodríguez-Luis, A.J., 2025. "A new kind of T-point in the Lorenz system with a different bifurcation set," Chaos, Solitons & Fractals, Elsevier, vol. 199(P1).
  • Handle: RePEc:eee:chsofr:v:199:y:2025:i:p1:s0960077925006824
    DOI: 10.1016/j.chaos.2025.116669
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077925006824
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.chaos.2025.116669?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.

    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:199:y:2025:i:p1:s0960077925006824. 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.

    We have no bibliographic references for this item. You can help adding them by using 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.