Centers of quadratic systems. (English) Zbl 0685.34024

Conti, Roberto

Ric. Mat. 36, Suppl., 117-126 (1987).

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