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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Main Authors:	Shugang Song, Jingjing Feng, Qinlong Wang
Format:	Article
Language:	English
Published:	Hindawi Limited 2014-01-01
Series:	Journal of Applied Mathematics
Online Access:	http://dx.doi.org/10.1155/2014/546243

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spelling	doaj-6cd86f2618524b7481122754dfd417b72020-11-24T21:03:00ZengHindawi LimitedJournal 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
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Shugang Song Jingjing Feng Qinlong Wang
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
author_facet	Shugang Song Jingjing Feng Qinlong Wang
author_sort	Shugang Song
title	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_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_sort	center conditions and bifurcation of limit cycles at the degenerate singularity of a three-dimensional system
publisher	Hindawi Limited
series	Journal of Applied Mathematics
issn	1110-757X 1687-0042
publishDate	2014-01-01
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.
url	http://dx.doi.org/10.1155/2014/546243
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The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System

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