## Centers of quadratic systems.(English)Zbl 0685.34024

A quadratic system in the autonomous system $$\dot x=P(x,y)$$, $$\dot y=Q(x,y)$$, where $P(x,y)=\sum^{2}_{j+\ell =0}a_{j,\ell}x^ jy^{\ell},\quad Q(x,y)=\sum^{2}_{j+\ell =0}b_{j,\ell}x^ jy^{\ell}$ are relatively prime real polynomials of degree at most two which are not both linear. A singular point S of the system is a center when there is a neighbourhood of S entirely covered by cycles. Let $$N_ S$$ be the union of the interior regions of the cycles surrounding S. The author shows that there are only five topologically different types of $$N_ S$$, and that there are only four possible combinations of $$N_ S$$ for a system.
Reviewer: P.Smith

### MSC:

 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations

### Keywords:

quadratic systems; centers; autonomous system; cycles