IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v177y2023ics0960077923011347.html
   My bibliography  Save this article

Finding nonlocal Lie symmetries algorithmically

Author

Listed:
  • Duarte, L.G.S.
  • da Mota, L.A.C.P.
  • Rocha, A.F.

Abstract

Here we present a new approach to compute symmetries of rational second order ordinary differential equations (rational 2ODEs). This method can compute Lie symmetries (point symmetries, dynamical symmetries and nonlocal symmetries) admitted by a rational 2ODE if it presents a Liouvillian first integral. The procedure is based on an idea arising from the formal equivalence between the total derivative operator and the vector field associated with the 2ODE over its solutions (Cartan vector field). Basically, from the formal representation of a Lie symmetry it is possible to extract information that allows to use this symmetry practically (in the 2ODE integration process) even in cases where the formal operation cannot be performed, i.e., in cases where the symmetry is nonlocal. Furthermore, when the 2ODE in question depends on parameters, the procedure allows an analysis that determines the regions of the parameter space in which the integrable cases are located. We finish by applying the method to the 2ODE that represents a forced modified Duffing–Van der Pol oscillator (which presents chaos for arbitrary values of the parameters) and determining a region of the parameter space that leads to integrable cases, some not previously known.

Suggested Citation

  • Duarte, L.G.S. & da Mota, L.A.C.P. & Rocha, A.F., 2023. "Finding nonlocal Lie symmetries algorithmically," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
    • Handle: RePEc:eee:chsofr:v:177:y:2023:i:c:s0960077923011347
    DOI: 10.1016/j.chaos.2023.114232
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077923011347
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2023.114232?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:177:y:2023:i:c:s0960077923011347. 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.