Nine Limit Cycles in a 5-Degree Polynomials Liénard System
In this article, we study the limit cycles in a generalized 5-degree Liénard system. The undamped system has a polycycle composed of a homoclinic loop and a heteroclinic loop. It is proved that the system can have 9 limit cycles near the boundaries of the period annulus of the undamped system. The m...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2020-01-01
|
| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2020/8584616 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832554076122906624 |
|---|---|
| author | Junning Cai Minzhi Wei Hongying Zhu |
| author_facet | Junning Cai Minzhi Wei Hongying Zhu |
| author_sort | Junning Cai |
| collection | DOAJ |
| description | In this article, we study the limit cycles in a generalized 5-degree Liénard system. The undamped system has a polycycle composed of a homoclinic loop and a heteroclinic loop. It is proved that the system can have 9 limit cycles near the boundaries of the period annulus of the undamped system. The main methods are based on homoclinic bifurcation and heteroclinic bifurcation by asymptotic expansions of Melnikov function near the singular loops. The result gives a relative larger lower bound on the number of limit cycles by Poincaré bifurcation for the generalized Liénard systems of degree five. |
| format | Article |
| id | doaj-art-3b571e39c38e48adbe2671bcc7748a3a |
| institution | Kabale University |
| issn | 1076-2787 1099-0526 |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-3b571e39c38e48adbe2671bcc7748a3a2025-02-03T05:52:24ZengWileyComplexity1076-27871099-05262020-01-01202010.1155/2020/85846168584616Nine Limit Cycles in a 5-Degree Polynomials Liénard SystemJunning Cai0Minzhi Wei1Hongying Zhu2Department of Information and Statistics, Guangxi University of Finance and Economics, Nanning, Guangxi 530003, ChinaDepartment of Information and Statistics, Guangxi University of Finance and Economics, Nanning, Guangxi 530003, ChinaDepartment of Information and Statistics, Guangxi University of Finance and Economics, Nanning, Guangxi 530003, ChinaIn this article, we study the limit cycles in a generalized 5-degree Liénard system. The undamped system has a polycycle composed of a homoclinic loop and a heteroclinic loop. It is proved that the system can have 9 limit cycles near the boundaries of the period annulus of the undamped system. The main methods are based on homoclinic bifurcation and heteroclinic bifurcation by asymptotic expansions of Melnikov function near the singular loops. The result gives a relative larger lower bound on the number of limit cycles by Poincaré bifurcation for the generalized Liénard systems of degree five.http://dx.doi.org/10.1155/2020/8584616 |
| spellingShingle | Junning Cai Minzhi Wei Hongying Zhu Nine Limit Cycles in a 5-Degree Polynomials Liénard System Complexity |
| title | Nine Limit Cycles in a 5-Degree Polynomials Liénard System |
| title_full | Nine Limit Cycles in a 5-Degree Polynomials Liénard System |
| title_fullStr | Nine Limit Cycles in a 5-Degree Polynomials Liénard System |
| title_full_unstemmed | Nine Limit Cycles in a 5-Degree Polynomials Liénard System |
| title_short | Nine Limit Cycles in a 5-Degree Polynomials Liénard System |
| title_sort | nine limit cycles in a 5 degree polynomials lienard system |
| url | http://dx.doi.org/10.1155/2020/8584616 |
| work_keys_str_mv | AT junningcai ninelimitcyclesina5degreepolynomialslienardsystem AT minzhiwei ninelimitcyclesina5degreepolynomialslienardsystem AT hongyingzhu ninelimitcyclesina5degreepolynomialslienardsystem |