On centers of type B of polynomial systems. (English) Zbl 0732.34027

Conti, Roberto

Arch. Math., Brno 26, No. 2-3, 93-100 (1990).

A planar system is considered: \(\dot x=X(x,y)\), \(\dot y=Y(x,y)\). Let S be a center and \(G_ s\) be a family of cycles surrounding S and no other singular point. Let int \(\gamma\) denote the region interior to \(\gamma\). Let \(N_ S=\cup_{\gamma \in G_ s}int \gamma\). A center is said to be of type B if \(\partial N_ S\) is the union of open orbits only.
Suppose that the planar system is polynomial of degree n. Then it is proved that a continuous band of cycles surrounding a center of type B is bounded by a number of orbits not greater than \(n+1\). On examples it is shown that such number can be n-1. It is conjectured that it cannot be greater than n-1. The same examples show that a system of degree n can have up to n centers of type B. It is conjectured that the number of such centers cannot be greater than n.

Reviewer: I.Ginchev (Varna)

MSC:

34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25	Periodic solutions to ordinary differential equations

Keywords:

polynomial systems; center; family of cycles; examples