Skip to main content
Log in

A Complement to the Uniqueness of the Limit Cycle of a Family of Systems with Homogeneous Components

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript


Consider the number of limit cycles of a family of systems with homogeneous components: \( {\dot{x}}=y, {\dot{y}}=-x^3+\alpha x^2y+y^3. \) We show that there is an \(\alpha ^*<0\) such that the system has exactly one limit cycle for \(\alpha \in (\alpha ^*,0),\) while no limit cycle for the else region. This completes a previous result and also gives a positive answer to the second part of Gasull’s 3rd problem listed in the paper (SeMA J 78(3):233–269, 2021). To obtain this result, we mainly analyse the behavior of the heteroclinic separatrices at infinity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others


  1. Cima, A., Gasukk, A., Mañosas, F.: Limit cycles for vector fields with homogeneous components. Appl. Math. 24, 281–287 (1997)

    MathSciNet  Google Scholar 

  2. Gasull, A.: Some open problems in low dimensional dynamical systems. SeMA J. 78(3), 233–269 (2021)

    Article  MathSciNet  Google Scholar 

  3. Gasull, A., Giacomini, H.: Upper bounds for the number of limit cycles through linear differential equations. Pac. J. Math. 226(2), 277–296 (2006)

    Article  MathSciNet  Google Scholar 

  4. Cima, A., Gasull, A., Mañosas, F.: Cyclicity of a family of vector fields. J. Math. Anal. Appl. 196(3), 921–937 (1995)

    Article  MathSciNet  Google Scholar 

  5. Perko, L.M.: Rotated vector fields. J. Differ. Equ. 103(1), 127–145 (1993)

    Article  MathSciNet  Google Scholar 

  6. Giacomini, H., Grau, M.: Transversal conics and the existence of limit cycles. J. Math. Anal. Appl. 428(1), 563–586 (2015)

    Article  MathSciNet  Google Scholar 

  7. Dumortier, F., Llibre, J., Artés, J.C.: Qualitative Theory of Planar Differential Systems, vol. 2. Springer, Berlin (2006)

    Google Scholar 

  8. McNabb, A.: Comparison theorems for differential equations. J. Math. Anal. Appl. 119(1–2), 417–428 (1986)

    Article  MathSciNet  Google Scholar 

Download references


This work is supported by the National Natural Science Foundation of China (No. 12171491). The authors would like to thank the editor Jaume Giné for handling the submission and the anonymous reviewers for their useful comments.

Author information

Authors and Affiliations



ZZ wrote the main manuscript text. All authors reviewed the manuscript.

Corresponding author

Correspondence to Ziwei Zhuang.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhuang, Z., Liu, C. A Complement to the Uniqueness of the Limit Cycle of a Family of Systems with Homogeneous Components. Qual. Theory Dyn. Syst. 23, 205 (2024).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: