Bautin’s theorem

There are at most three limit cycles which can appear in the following quadratic system

	$\\displaystyle\\dot{x}=p(x,y)$	$\\displaystyle=$	$\\displaystyle\\sum_{i+j=0}^{2}a_{ij}x^{i}y^{j}$
	$\\displaystyle\\dot{y}=q(x,y)$	$\\displaystyle=$	$\\displaystyle\\sum_{i+j=0}^{2}b_{ij}x^{i}y^{j}$

from a singular point, if its type is either a focus or a center.

In 1939 N.N. Bautin claimed the above result and in 1952 submitted the proof [BNN1]. [GAV]

References

GAV Gaiko, A., Valery: Global Bifurcation Theory and Hilbert’s Sixteenth Problem. Kluwer Academic Publishers, London, 2003.
BNN1 Bautin, N.N.: On the number of limit cycles appearing from an equilibrium point of the focus or center type under varying coefficients. Matem. SB., 30:181-196, 1952. (written in Russian)
BNN2 Bautin, N.N.: On the number of limit cycles appearing from an equilibrium point of the focus or center type under varying coefficients. Translation of the American Mathematical Society, 100, 1954.