The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System
We investigate multiple limit cycles bifurcation and center-focus problem of the degenerate equilibrium for a three-dimensional system. By applying the method of symbolic computation, we obtain the first four quasi-Lyapunov constants. It is proved that the system can generate 3 small limit cycles fr...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2014/546243 |
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| author | Shugang Song Jingjing Feng Qinlong Wang |
| author_facet | Shugang Song Jingjing Feng Qinlong Wang |
| author_sort | Shugang Song |
| collection | DOAJ |
| description | We investigate multiple limit cycles bifurcation and center-focus problem of the degenerate equilibrium for a three-dimensional system. By applying the method of symbolic computation, we obtain the first four quasi-Lyapunov constants. It is proved that the system can generate 3 small limit cycles from nilpotent critical point on center manifold. Furthermore, the center conditions are found and as weak foci the highest order is proved to be the fourth; thus we obtain at most 3 small limit cycles from the origin via local bifurcation. To our knowledge, it is the first example of multiple limit cycles bifurcating from a nilpotent singularity for the flow of a high-dimensional system restricted to the center manifold. |
| format | Article |
| id | doaj-art-6cd86f2618524b7481122754dfd417b7 |
| institution | OA Journals |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-6cd86f2618524b7481122754dfd417b72025-08-20T02:24:03ZengWileyJournal of Applied Mathematics1110-757X1687-00422014-01-01201410.1155/2014/546243546243The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional SystemShugang Song0Jingjing Feng1Qinlong Wang2School of Information and Mathematics, Yangtze University, Jingzhou 434023, ChinaSchool of Information and Mathematics, Yangtze University, Jingzhou 434023, ChinaSchool of Science, Hezhou University, Hezhou 542899, ChinaWe investigate multiple limit cycles bifurcation and center-focus problem of the degenerate equilibrium for a three-dimensional system. By applying the method of symbolic computation, we obtain the first four quasi-Lyapunov constants. It is proved that the system can generate 3 small limit cycles from nilpotent critical point on center manifold. Furthermore, the center conditions are found and as weak foci the highest order is proved to be the fourth; thus we obtain at most 3 small limit cycles from the origin via local bifurcation. To our knowledge, it is the first example of multiple limit cycles bifurcating from a nilpotent singularity for the flow of a high-dimensional system restricted to the center manifold.http://dx.doi.org/10.1155/2014/546243 |
| spellingShingle | Shugang Song Jingjing Feng Qinlong Wang The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System Journal of Applied Mathematics |
| title | The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System |
| title_full | The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System |
| title_fullStr | The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System |
| title_full_unstemmed | The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System |
| title_short | The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System |
| title_sort | center conditions and bifurcation of limit cycles at the degenerate singularity of a three dimensional system |
| url | http://dx.doi.org/10.1155/2014/546243 |
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