Uniqueness of Limit Cycles for a Class of Cubic Systems with Two Invariant Straight Lines
A class of cubic systems with two invariant straight lines dx/dt=y(1-x2), dy/dt=-x+δy+nx2+mxy+ly2+bxy2. is studied. It is obtained that the focal quantities of O(0,0) are, W0=δ; if W0=0, then W1=m(n+l); if W0=W1=0, then W2=−nm(b+1); if W0=W1=W2=0, then O is a center, and it has been proved that the...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2010-01-01
|
| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2010/737068 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832564768302432256 |
|---|---|
| author | Xiangdong Xie Fengde Chen Qingyi Zhan |
| author_facet | Xiangdong Xie Fengde Chen Qingyi Zhan |
| author_sort | Xiangdong Xie |
| collection | DOAJ |
| description | A class of cubic systems with two invariant straight lines dx/dt=y(1-x2), dy/dt=-x+δy+nx2+mxy+ly2+bxy2. is studied. It is obtained that the focal quantities of O(0,0) are, W0=δ; if W0=0, then W1=m(n+l); if W0=W1=0, then W2=−nm(b+1); if W0=W1=W2=0, then O is a center, and it has been proved that the above mentioned cubic system has at most one limit cycle surrounding weak focal O(0,0). This paper also aims to solve the remaining issues in the work of Zheng and Xie (2009). |
| format | Article |
| id | doaj-art-98575f873a1048628842449c1814a3ec |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-98575f873a1048628842449c1814a3ec2025-02-03T01:10:15ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2010-01-01201010.1155/2010/737068737068Uniqueness of Limit Cycles for a Class of Cubic Systems with Two Invariant Straight LinesXiangdong Xie0Fengde Chen1Qingyi Zhan2Department of Mathematics, Ningde Normal University, Ningde, Fujian 352100, ChinaSchool of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, ChinaCollege of Computer and Information Science, Fujian Agriculture and Forestry University, Fuzhou, Fujian 350002, ChinaA class of cubic systems with two invariant straight lines dx/dt=y(1-x2), dy/dt=-x+δy+nx2+mxy+ly2+bxy2. is studied. It is obtained that the focal quantities of O(0,0) are, W0=δ; if W0=0, then W1=m(n+l); if W0=W1=0, then W2=−nm(b+1); if W0=W1=W2=0, then O is a center, and it has been proved that the above mentioned cubic system has at most one limit cycle surrounding weak focal O(0,0). This paper also aims to solve the remaining issues in the work of Zheng and Xie (2009).http://dx.doi.org/10.1155/2010/737068 |
| spellingShingle | Xiangdong Xie Fengde Chen Qingyi Zhan Uniqueness of Limit Cycles for a Class of Cubic Systems with Two Invariant Straight Lines Discrete Dynamics in Nature and Society |
| title | Uniqueness of Limit Cycles for a Class of Cubic
Systems with Two Invariant Straight Lines |
| title_full | Uniqueness of Limit Cycles for a Class of Cubic
Systems with Two Invariant Straight Lines |
| title_fullStr | Uniqueness of Limit Cycles for a Class of Cubic
Systems with Two Invariant Straight Lines |
| title_full_unstemmed | Uniqueness of Limit Cycles for a Class of Cubic
Systems with Two Invariant Straight Lines |
| title_short | Uniqueness of Limit Cycles for a Class of Cubic
Systems with Two Invariant Straight Lines |
| title_sort | uniqueness of limit cycles for a class of cubic systems with two invariant straight lines |
| url | http://dx.doi.org/10.1155/2010/737068 |
| work_keys_str_mv | AT xiangdongxie uniquenessoflimitcyclesforaclassofcubicsystemswithtwoinvariantstraightlines AT fengdechen uniquenessoflimitcyclesforaclassofcubicsystemswithtwoinvariantstraightlines AT qingyizhan uniquenessoflimitcyclesforaclassofcubicsystemswithtwoinvariantstraightlines |