Limit Cycles and Analytic Centers for a Family of 4n-1 Degree Systems with Generalized Nilpotent Singularities
With the aid of computer algebra system Mathematica 8.0 and by the integral factor method, for a family of generalized nilpotent systems, we first compute the first several quasi-Lyapunov constants, by vanishing them and rigorous proof, and then we get sufficient and necessary conditions under whic...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2015-01-01
|
| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2015/859015 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832562609068441600 |
|---|---|
| author | Yusen Wu Cui Zhang Changjin Xu |
| author_facet | Yusen Wu Cui Zhang Changjin Xu |
| author_sort | Yusen Wu |
| collection | DOAJ |
| description | With the aid of computer algebra system Mathematica 8.0 and by the integral factor method, for a family of generalized nilpotent systems, we first compute the first several quasi-Lyapunov constants, by vanishing them and rigorous proof, and then we get sufficient and necessary conditions under which the systems admit analytic centers at the origin. In addition, we present that seven amplitude limit cycles can be created from the origin. As an example, we give a concrete system with seven limit cycles via parameter perturbations to illustrate our conclusion. An interesting phenomenon is that the exponent parameter n controls the singular point type of the studied system. The main results generalize and improve the previously known results in Pan. |
| format | Article |
| id | doaj-art-6929daabd6f248839de8b9c463ce6295 |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-6929daabd6f248839de8b9c463ce62952025-02-03T01:22:12ZengWileyJournal of Function Spaces2314-88962314-88882015-01-01201510.1155/2015/859015859015Limit Cycles and Analytic Centers for a Family of 4n-1 Degree Systems with Generalized Nilpotent SingularitiesYusen Wu0Cui Zhang1Changjin Xu2School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, Henan 471023, ChinaCollege of Mathematical Science, Luoyang Normal University, Luoyang, Henan 471022, ChinaGuizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang, Guizhou 550004, ChinaWith the aid of computer algebra system Mathematica 8.0 and by the integral factor method, for a family of generalized nilpotent systems, we first compute the first several quasi-Lyapunov constants, by vanishing them and rigorous proof, and then we get sufficient and necessary conditions under which the systems admit analytic centers at the origin. In addition, we present that seven amplitude limit cycles can be created from the origin. As an example, we give a concrete system with seven limit cycles via parameter perturbations to illustrate our conclusion. An interesting phenomenon is that the exponent parameter n controls the singular point type of the studied system. The main results generalize and improve the previously known results in Pan.http://dx.doi.org/10.1155/2015/859015 |
| spellingShingle | Yusen Wu Cui Zhang Changjin Xu Limit Cycles and Analytic Centers for a Family of 4n-1 Degree Systems with Generalized Nilpotent Singularities Journal of Function Spaces |
| title | Limit Cycles and Analytic Centers for a Family of 4n-1 Degree Systems with Generalized Nilpotent Singularities |
| title_full | Limit Cycles and Analytic Centers for a Family of 4n-1 Degree Systems with Generalized Nilpotent Singularities |
| title_fullStr | Limit Cycles and Analytic Centers for a Family of 4n-1 Degree Systems with Generalized Nilpotent Singularities |
| title_full_unstemmed | Limit Cycles and Analytic Centers for a Family of 4n-1 Degree Systems with Generalized Nilpotent Singularities |
| title_short | Limit Cycles and Analytic Centers for a Family of 4n-1 Degree Systems with Generalized Nilpotent Singularities |
| title_sort | limit cycles and analytic centers for a family of 4n 1 degree systems with generalized nilpotent singularities |
| url | http://dx.doi.org/10.1155/2015/859015 |
| work_keys_str_mv | AT yusenwu limitcyclesandanalyticcentersforafamilyof4n1degreesystemswithgeneralizednilpotentsingularities AT cuizhang limitcyclesandanalyticcentersforafamilyof4n1degreesystemswithgeneralizednilpotentsingularities AT changjinxu limitcyclesandanalyticcentersforafamilyof4n1degreesystemswithgeneralizednilpotentsingularities |