Journal of Applied Mathematics

The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System

Shugang Song, Jingjing Feng, and Qinlong Wang

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Abstract

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.

Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 546243, 8 pages.

Dates
First available in Project Euclid: 2 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.jam/1425305889

Digital Object Identifier
doi:10.1155/2014/546243

Mathematical Reviews number (MathSciNet)
MR3232919

Citation

Song, Shugang; Feng, Jingjing; Wang, Qinlong. The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System. J. Appl. Math. 2014 (2014), Article ID 546243, 8 pages. doi:10.1155/2014/546243. https://projecteuclid.org/euclid.jam/1425305889


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