## 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.
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
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