Sufficient conditions for the existence of at least \(n\) or exactly \(n\) limit cycles for the Liénard differential systems. (English) Zbl 1131.34026
The authors consider the planar autonomous system
\[ \dot{x}=y-F(x),\quad \dot{y}=-g(x) \]
corresponding to the Liénard equation. They give conditions for the existence of not less than \(n\) limit cycles (or exactly \(n\) limit cycles).
\[ \dot{x}=y-F(x),\quad \dot{y}=-g(x) \]
corresponding to the Liénard equation. They give conditions for the existence of not less than \(n\) limit cycles (or exactly \(n\) limit cycles).
Reviewer: Sergei Yu. Pilyugin (St. Petersburg)
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
References:
| [1] | Alshom, P., Existence of limit cycles for generalized Liénard equations, J. Math. Anal. Appl., 171, 242-255 (1992) · Zbl 0763.34016 |
| [2] | Blows, T. R.; Lloyd, N. G., The number of small-amplitude limit cycles of Liénard equations, Math. Proc. Cambridge Philos. Soc., 95, 359-366 (1984) · Zbl 0532.34022 |
| [3] | Chen, Xiudong, Properties of characteristic functions and existence of limit cycles of Liénard’s equation, Chinese Ann. Math., 4B, 2, 207-215 (1983) · Zbl 0511.34022 |
| [4] | Chen, Xiudong; Chen, Yong, A sufficient condition for Liénard’s equation that has at most \(n\) limit cycles, J. Math. Res. Exposition, 23, 333-338 (2003), (in Chinese) · Zbl 1116.34314 |
| [5] | Dumortier, F.; Llibre, J.; Artés, J. C., Qualitative Theory of Planar Differential Systems, Universitext (2006), Springer · Zbl 1110.34002 |
| [6] | Dumortier, F.; Panazzolo, D.; Roussarie, R., More limit cycles than expected in Liénard systems, Proc. Amer. Math. Soc., 135, 1895-1904 (2007) · Zbl 1130.34018 |
| [7] | Écalle, J., Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac (1992), Hermann · Zbl 1241.34003 |
| [8] | Hilbert, D., Mathematische Probleme, (Lecture, Second Internat. Congr. Math.. Lecture, Second Internat. Congr. Math., Paris, 1900. Lecture, Second Internat. Congr. Math.. Lecture, Second Internat. Congr. Math., Paris, 1900, Nachr. Ges. Wiss. Göttingen Math. Phys. KL. (1900)). (Lecture, Second Internat. Congr. Math.. Lecture, Second Internat. Congr. Math., Paris, 1900. Lecture, Second Internat. Congr. Math.. Lecture, Second Internat. Congr. Math., Paris, 1900, Nachr. Ges. Wiss. Göttingen Math. Phys. KL. (1900)), Bull. Amer. Math. Soc., 8, 437-479 (1902), English transl.: · JFM 33.0976.07 |
| [9] | Ilyashenko, Yu., Finiteness Theorems for Limit Cycles, Transl. Math. Monogr., vol. 94 (1991), Amer. Math. Soc. · Zbl 0743.34036 |
| [10] | Ilyashenko, Yu.; Panov, A., Some upper estimates of the number of limit cycles of planar vector fields with applications to Liénard equations, Mosc. Math. J., 1, 583-599 (2001) · Zbl 1008.34027 |
| [11] | Lins, A.; de Melo, W.; Pugh, C. C., On Liénard’s equation, (Lecture Notes in Math., vol. 597 (1977), Springer: Springer Berlin), 335-357 · Zbl 0362.34022 |
| [12] | Llibre, J.; Rodríguez, G., Configurations of limit cycles and planar polynomial vector fields, J. Differential Equations, 198, 374-380 (2004) · Zbl 1055.34061 |
| [13] | López, J. L.; López-Ruiz, R., The limit cycles of Liénard equations in the strongly nonlinear regime, Chaos Solitons Fractals, 11, 747-756 (2000) · Zbl 0954.34021 |
| [14] | Odani, K., Existence of exactly \(N\) periodic solutions for Liénard systems, Funkcial. Ekvac., 39, 217-234 (1996) · Zbl 0864.34032 |
| [15] | Smale, S., Mathematical problems for the next century, Math. Intelligencer, 20, 7-15 (1998) · Zbl 0947.01011 |
| [16] | Xianwu, Zeng; Zhifen, Zhang; Suzhi, Gao, On the uniqueness of the generalized Liénard equation, Bull. London Math. Soc., 26, 213-247 (1994) · Zbl 0805.34031 |
| [17] | Dongmei Xio, Zhifen Zhang, On the uniqueness of limit cycles for the generalized Liénard systems, J. Math. Anal. Appl. (2007), in press; Dongmei Xio, Zhifen Zhang, On the uniqueness of limit cycles for the generalized Liénard systems, J. Math. Anal. Appl. (2007), in press |
| [18] | Xiao, Dongmei; Zhang, Zhifen, Liénard’s equation with sufficient conditions and constructing method for the existence of at least \(n\) limit cycles, J. North-Eastern Normal University (Natural Sciences Edition), 1, 1-8 (1983), (in Chinese) · Zbl 0519.34023 |
| [19] | Zeng, Xianwu; Zhang, Zhigen; Gao, Suzhi, On the uniqueness of the limit cycle of the generalized Liénard equation, Bull. London Math. Soc., 26, 213-247 (1994) · Zbl 0805.34031 |
| [20] | Zhang, Zhifen; Chiming, Ho, On the sufficient conditions for the existence of at most \(n\) limit cycles of Liénard equation, Acta Math. Sinica, 25, 5, 585-594 (1982), (in Chinese) · Zbl 0525.34024 |
| [21] | Zhifen, Zhang; Tongren, Ding; Wenzao, Huang; Zhenxi, Dong, Qualitative Theory of Differential Equations, Transl. Math. Monogr., vol. 101 (1992), Amer. Math. Soc.: Amer. Math. Soc. Providence · Zbl 0779.34001 |
| [22] | Zuppa, C., Order of cyclicity of the singular point of Liénard’s polynomial vector fields, Bol. Soc. Brasil. Mat., 12, 105-111 (1981) · Zbl 0577.34023 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
