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
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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