A kind of bifurcation of limit cycle from nilpotent critical point

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In this paper, an interesting and new bifurcation phenomenon that limit cycles could be bifurcated from nilpotent node (focus) by changing its stability was investigated. It is different from lowing its multiplicity in order to get limit cycles. We prove that n²+n-1 limit cycles could be bifurcated by this way for 2n+1 degree system. Moreover, this upper bound could be reached. At last, we give two examples to show that N(3)=1 and N(5)=5.