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

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Shugang Song ◽  
Jingjing Feng ◽  
Qinlong Wang
Keyword(s):  
Limit Cycles ◽  
Center Manifold ◽  
Three Dimensional ◽  
High Dimensional ◽  
Dimensional System ◽  
Bifurcation Of Limit Cycles ◽  
Center Conditions ◽  
Nilpotent Critical Point ◽  
Three Dimensional System ◽  
Lyapunov Constants

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.

scholarly journals Fourteen Limit Cycles in a Seven-Degree Nilpotent System

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Wentao Huang ◽  
Ting Chen ◽  
Tianlong Gu
Keyword(s):  
Critical Point ◽  
Computer Algebra ◽  
Limit Cycles ◽  
Computer Algebra System ◽  
Degree Polynomial ◽  
Bifurcation Of Limit Cycles ◽  
Fine Focus ◽  
Center Conditions ◽  
Nilpotent Critical Point ◽  
Lyapunov Constants

Center conditions and the bifurcation of limit cycles for a seven-degree polynomial differential system in which the origin is a nilpotent critical point are studied. Using the computer algebra system Mathematica, the first 14 quasi-Lyapunov constants of the origin are obtained, and then the conditions for the origin to be a center and the 14th-order fine focus are derived, respectively. Finally, we prove that the system has 14 limit cycles bifurcated from the origin under a small perturbation. As far as we know, this is the first example of a seven-degree system with 14 limit cycles bifurcated from a nilpotent critical point.

scholarly journals Limit Cycles and Integrability in a Class of Systems with High-Order Nilpotent Critical Points

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Feng Li ◽  
Jianlong Qiu
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Limit Cycles ◽  
High Order ◽  
Bifurcation Of Limit Cycles ◽  
Polynomial Differential Systems ◽  
Sufficient And Necessary Conditions ◽  
Center Conditions ◽  
Nilpotent Critical Point ◽  
Lyapunov Constants

A class of polynomial differential systems with high-order nilpotent critical points are investigated in this paper. Those systems could be changed into systems with an element critical point. The center conditions and bifurcation of limit cycles could be obtained by classical methods. Finally, an example was given; with the help of computer algebra system MATHEMATICA, the first 5 Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact that there exist 5 small amplitude limit cycles created from the high-order nilpotent critical point is also proved.

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