IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i6p885-d366090.html
   My bibliography  Save this article

Rational Limit Cycles on Abel Polynomial Equations

Author

Listed:
  • Claudia Valls

    (Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal)

Abstract

In this paper we deal with Abel equations of the form d y / d x = A 1 ( x ) y + A 2 ( x ) y 2 + A 3 ( x ) y 3 , where A 1 ( x ) , A 2 ( x ) and A 3 ( x ) are real polynomials and A 3 ≢ 0 . We prove that these Abel equations can have at most two rational (non-polynomial) limit cycles when A 1 ≢ 0 and three rational (non-polynomial) limit cycles when A 1 ≡ 0 . Moreover, we show that these upper bounds are sharp. We show that the general Abel equations can always be reduced to this one.

Suggested Citation

  • Claudia Valls, 2020. "Rational Limit Cycles on Abel Polynomial Equations," Mathematics, MDPI, vol. 8(6), pages 1-15, June.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:6:p:885-:d:366090
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/6/885/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/6/885/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Liu, Changjian & Li, Chunhui & Wang, Xishun & Wu, Junqiao, 2018. "On the rational limit cycles of Abel equations," Chaos, Solitons & Fractals, Elsevier, vol. 110(C), pages 28-32.
    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. Claudia Valls, 2020. "On the Polynomial Solutions and Limit Cycles of Some Generalized Polynomial Ordinary Differential Equations," Mathematics, MDPI, vol. 8(7), pages 1-11, July.

    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. Claudia Valls, 2020. "On the Polynomial Solutions and Limit Cycles of Some Generalized Polynomial Ordinary Differential Equations," Mathematics, MDPI, vol. 8(7), pages 1-11, July.
    2. Llibre, Jaume & Valls, Claudia, 2021. "Phase portraits of the complex Abel polynomial differential systems," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).

    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:gam:jmathe:v:8:y:2020:i:6:p:885-:d:366090. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.