Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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Global dynamics of a Wilson polynomial Liénard equation
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by Haibo Chen and Hebai Chen
Proc. Amer. Math. Soc. 148 (2020), 4769-4780
DOI: https://doi.org/10.1090/proc/15074
Published electronically: July 30, 2020

Abstract:

Gasull and Sabatini in [Ann. Mat. Pura Appl. 198 (2019), pp. 1985–2006] studied limit cycles of a Liénard system which has a fixed invariant curve, i.e., a Wilson polynomial Liénard system. The Liénard system can be changed into $\dot x=y-(x^2-1)(x^3-bx), ~ \dot y=-x(1+y(x^3-bx))$. For $b\leq 0.7$ and $b\geq 0.76$, limit cycles of the system are studied completely. But for $0.7<b<0.76$, the exact number of limit cycles is still unknown, and Gasull and Sabatini conjectured that the exact number of limit cycles is two (including multiplicities). In this paper, we give a positive answer to this conjecture and study all bifurcations of the system. Finally, we show the expanding of the moving limit cycle as $b>0$ increases and give all global phase portraits on the Poincaré disk of the system completely.
References
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Bibliographic Information
  • Haibo Chen
  • Affiliation: School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, People’s Republic of China
  • Email: math_chb@csu.edu.cn
  • Hebai Chen
  • Affiliation: School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, People’s Republic of China
  • MR Author ID: 1112845
  • Email: chen_hebai@csu.edu.cn
  • Received by editor(s): January 12, 2020
  • Received by editor(s) in revised form: February 2, 2020
  • Published electronically: July 30, 2020
  • Additional Notes: The second author is the corresponding author.
    This paper was supported by the National Natural Science Foundation of China (Nos. 11801079, 11671403).
  • Communicated by: Wenxian Shen
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 4769-4780
  • MSC (2010): Primary 34C07, 34C23, 34C25, 37C27
  • DOI: https://doi.org/10.1090/proc/15074
  • MathSciNet review: 4143393