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A class of polynomial differential systems with high-order nil potent 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 nil potent critical point is also proved.
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