Skip to main content
SpringerLink
Log in

Existence and uniqueness of limit cycles in planar system of Liénard type

  • Published:
Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Abstract

This paper is a study of one of the most beautiful phenomena in dynamical systems limit cycle. In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory having the property that at least one other trajectory spirals into it as time approaches either positive or negative infinity. In this paper, existence and uniqueness of limit cycles for a generalized Liénard system will be studied. Moreover, some examples will be presented to illustrate our results.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Carletti, T.: Uniqueness of limit cycles for a class of planar vector fields. Qual. Theory Dyn. Syst. 6, 31–43 (2005)

    Article  MathSciNet  Google Scholar 

  2. Carletti, T., Villari, G.: A note on existence and uniqueness of limit cycles for Liénard systems. J. Math. Anal. Appl. 307, 763–773 (2005)

    Article  MathSciNet  Google Scholar 

  3. Da-zheng, D.: On the existence of limit cycles for the Liénard’s equation. Acta. Math. Appl. Sinica 7, 166–174 (1984). (Chinese)

    MathSciNet  Google Scholar 

  4. Dulac, H.: Sur les cycles limites. Bull. Soc. Math. France 51, 45–188 (1923)

    Article  MathSciNet  Google Scholar 

  5. Filippov, A.F.: A sufficient condition for the existence of a stable limit cycles for a second order equation. Mat. Sb. 30, 171–180 (1952). (in Russian)

    Google Scholar 

  6. Hayashi, M.: On the uniqueness of the closed orbit of the Linard system. Math. Japon. 46, 371–376 (1997)

    MathSciNet  MATH  Google Scholar 

  7. Hayashi, M., Tsukada, M.: A uniqueness theorem on the Linard system with a non- hyperbolic equilibrium point. Dyn. Syst. Appl. 9, 99–108 (2000)

    MATH  Google Scholar 

  8. Kooij, R.E., Jianhua, S.: A note on uniqueness of limit cycles in a Liénard-type system. J. Math. Anal. Appl. 208, 260–276 (1997)

    Article  MathSciNet  Google Scholar 

  9. Lefschetz, S.: Differential Equations: Geometric Theory. Dover- Publications Inc, New York (1977). (reprinting of the second edition)

    MATH  Google Scholar 

  10. Levinson, N., Smith, O.K.: A general equation for relaxation oscillations. Duke Math. J. 9, 382–403 (1942)

    Article  MathSciNet  Google Scholar 

  11. Liénard, A.: Étude des oscillations entretenues. Revue générale de lElectricité 23, 901–912 (1928)

    Google Scholar 

  12. Sabatini, M., Villari, G.: Limit cycle uniqueness for a class of planar dynamical systems. Appl. Math. Lett. 19, 1180–1184 (2006)

    Article  MathSciNet  Google Scholar 

  13. Sansone, G.: Sopra l’equazione di A. Lienard delle oscillazioni de rilassamento. Ann. Mat. Pure Appl. 28, 153–181 (1949)

    Article  MathSciNet  Google Scholar 

  14. Van der Pol, B.: On “relaxation-oscillations”. Lond. Edinb. Dublin Philos. Mag. J. Sci. Ser. 7(2), 978–992 (1926)

    Google Scholar 

  15. Villari, G.: Some remarks on the uniqueness of the periodic solutions for Linard’s equation. Boll. Un. Mat. Ital. C 6(4), 173–182 (1985)

    MathSciNet  MATH  Google Scholar 

  16. Villari, G., Zanolin, F.: On the uniqueness of the limit cycle for the Linard equation, via comparison method for the energy level curves. Dyn. Syst. Appl. 25, 321–334 (2016)

    MATH  Google Scholar 

  17. Villari, G., Zanolin, F.: On the uniqueness of the limit cycle for the Liénard equation with \(f(x)\) not sign-definite. Appl. Math. Lett. 76, 208–214 (2018)

    Article  MathSciNet  Google Scholar 

  18. Xiao, D., Zhang, Z.: On the uniqueness and nonexistence of limit cycles for predator-prey systems. Nonlinearity 16, 1185–1201 (2003)

    Article  MathSciNet  Google Scholar 

  19. Zhang, Z.: Proof of the uniqueness theorem of limit cycles of generalized Liénard equations. Appl. Anal. 23, 63–76 (1986)

    Article  MathSciNet  Google Scholar 

  20. Zhang, Z., Ding, T., Huang, W., Dong, Z.: Qualitative Theory of Differential Equations. Translation of Mathematical Monographs, vol. 102. AMS, Providence (1992)

    Google Scholar 

  21. Zhang, P., Cai, S.: On the uniqueness of limit cycle of equation \({\dot{x}} = \varPhi (y)-F(x), {\dot{y}} =-g(x),\). J. Zhejiang Univ. 24, 443–448 (1990)

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank anonymous referees for their valuable comments that improved the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematics, University of Tabriz, Tabriz, Iran

    Tohid Kasbi

  2. Department of Mathematics, Faculty of Basic Sciences, Azarbaijan Shahid Madani University, Tabriz, Iran

    Vahid Roomi

Authors
  1. Tohid Kasbi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vahid Roomi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vahid Roomi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kasbi, T., Roomi, V. Existence and uniqueness of limit cycles in planar system of Liénard type. Rend. Circ. Mat. Palermo, II. Ser 67, 547–556 (2018). https://doi.org/10.1007/s12215-018-0333-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-018-0333-2

Keywords

Mathematics Subject Classification

Search

Navigation