A kind of bifurcation of limit cycle from a nilpotent critical point

In this paper, an interesting and new bifurcation phenomenon that limit cycles could be bifurcated from nilpotent node (focus) by changing its stability is investigated. It is different from lowing its multiplicity in order to get limit cycles. We prove that $n^2+n-1$ limit cycles could be bifurcated by this way for $2n+1$ degree systems. Moreover, this upper bound could be reached. At last, we give two examples to show that $N(3)=1$ and $N(5)=5$ respectively. Here, $N(n)$ denotes the number of small-amplitude limit cycles around a nilpotent node (focus) with $n$ being the degree of polynomials in the vector field.