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

Song Shu-gang | Zendy; Jingjing Feng | Zendy; Qinlong Wang | Zendy

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

Author(s) -

Song Shu-gang,

Jingjing Feng,

Qinlong Wang

Publication year - 2014

Publication title -

journal of applied mathematics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.307

H-Index - 43

eISSN - 1687-0042

pISSN - 1110-757X

DOI - 10.1155/2014/546243

Subject(s) - center manifold , mathematics , singularity , bifurcation , limit (mathematics) , degenerate energy levels , center (category theory) , nilpotent , infinite period bifurcation , mathematical analysis , singular point of a curve , limit cycle , manifold (fluid mechanics) , saddle node bifurcation , pure mathematics , hopf bifurcation , physics , nonlinear system , mechanical engineering , chemistry , quantum mechanics , engineering , crystallography

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

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