# 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.
Title Bautin’s theorem BautinsTheorem 2013-03-22 14:28:46 2013-03-22 14:28:46 Daume (40) Daume (40) 5 Daume (40) Theorem msc 34C07