The center-focus problem and reversibility. (English) Zbl 0994.34020

Teixeira, Marco Antonio; Yang, Jiazhong

J. Differ. Equations 174, No. 1, 237-251 (2001).

The reversibility is an interesting concept useful in the qualitative theory of differential equations that implies certain geometric properties. For instance, if an orbit of a reversible vector field meets the fixed-point set at two distinct points then it is necessarily a symmetric periodic orbit.
The paper is concerned with the following problem for a class of two-dimensional systems having a center at the singular point: if a system possesses certain symmetries, what can be said about its reversibility? Sufficient and necessary conditions for the analytic planar system to have a center are given and the reversibility of certain classes of polynomial vector fields is discussed.

Reviewer: Yuri V.Rogovchenko (Famagusta)

MSC:

34C05

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

Keywords:

center; focus; involution; reversibility; analytic vector field; monodromy conditions; normal forms

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