In this paper, a class of polynomial differential system with high-order critical point are investigated. The system could be changed into a system with a 3-order nilpotent critical point. Finally, an example was given, with the help of computer algebra system MATHEMATICA, the first three quasi-Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact that there three small amplitude limit cycles created from the 3-order nilpotent critical point is also proved.
High-order critical point Nilpotent critical point Center Focus Bifurcation of limit cycle
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This research is partially supported by the National Nature Science Foundation of China (11201211, 61273012) and Nature Science Foundation of Shan-dong Province (ZR2012AL04).
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