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

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.

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)