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An Improved Criterion for the Unique Existence of the Limit Cycle of a Liénard-type System with One Parameter

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Abstract

There are various types in generalized Liénard systems. In this paper, we treat a Liénard-type system in their paper. Though the existence of homoclinic orbits was discussed in their paper, our aim is to present a tool for the unique existence of limit cycles of the system with one parameter. It is an improvement of the criterion in Hayashi (Adv Dyn Syst Appl 14:179–187, 2019). As applications, the examples in the above paper of Hayashi or Villari and Zanolin (Dyn Syst Appl 25:321–334, 2016) are investigated under the more small parameter for the uniqueness of the limit cycle.

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References

  1. Aghajani, A., Moradifam, A.: The generalized Liénard equations. Glasgow Math. J. 51, 605–617 (2009)

    Article  MathSciNet  Google Scholar 

  2. 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 

  3. Cioni, M., Villari, G.: An extention of Dragilev’s theorem for the existence of periodic solution of the Liénard equation. Nonlinear Anal. 20, 345–351 (2015)

    MATH  Google Scholar 

  4. Gasull, A., Guillamon, A.: Non-existence, uniqueness of limit cycles and center problem in a system that includes predator-prey systems and generalized Liénard equations. Differ. Equ. Dyn. Syst. 3, 336–345 (1995)

    MATH  Google Scholar 

  5. Graef, J.: On the generalized Liénard equation with negative damping. J. Differ. Equ. 12, 33–74 (1972)

    Article  Google Scholar 

  6. Hayashi, M.: On the uniqueness of the closed orbit of the Liénard system. Math. Jpn. 47, 1–6 (1997)

    Google Scholar 

  7. Hayashi, M.: A criterion for the unique existence of the limit cycle of a Liénard-type system with one parameter. Adv. Dyn. Syst. Appl. 14(2), 179–187 (2019)

    MathSciNet  Google Scholar 

  8. Hayashi, M., Tsukada, M.: A uniqueness theorem on the Liénard systems with a non-hyperbolic equilibrium point. Dyn. Syst. Appl. 9, 98–108 (2000)

    MATH  Google Scholar 

  9. Hayashi, M., Villari, G., Zanolin, F.: On the uniqueness of limit cycle for certain Liénard systems without symmetry. Elect. J. Qual. Theory Differ. Equ. 55, 1–10 (2018)

    MATH  Google Scholar 

  10. Xun-Cheng, Huang: Uniqueness of limit cycles of generalized Liénard systems and predator-prey systems. J. Phys. A Math. Gen. 21, L681–L685 (1988)

    Article  Google Scholar 

  11. Lefschetz, S.: Differential Equations: Geometric Theory, 2nd edn. Dover Publications, New York (1977)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  14. Xian, D., Zhang, Z.: On the uniqueness and non-existence of limit cycles for predator-prey systems. Nonlinearity 16, 1185–1201 (2003)

    Article  MathSciNet  Google Scholar 

  15. Zhang, D., Yan, P.: On the uniqueness of limit cycles in a generalized Liénard system. Qual. Theory Dyn. Syst. 18(3), 1191–1199 (2019)

    Article  MathSciNet  Google Scholar 

  16. Zhi-fen, Z., Tong-ren, D., Wen-zao, H., Zhen-xi, D.: Qualitative Theory of Differential Equations. Translations of Mathematical Monographs in AMS 102, (1992)

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Acknowledgements

The author thanks the referee for the useful comments and advice.

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  1. Department of Mathematics, College of Science and Technology, Nihon University, Funabashi, Chiba, 274-8501, Japan

    Makoto Hayashi

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  1. Makoto Hayashi
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Correspondence to Makoto Hayashi.

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Hayashi, M. An Improved Criterion for the Unique Existence of the Limit Cycle of a Liénard-type System with One Parameter. Qual. Theory Dyn. Syst. 20, 25 (2021). https://doi.org/10.1007/s12346-021-00463-z

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