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Centers of quasi-homogeneous polynomial planar systems. (English) Zbl 1238.34052

Summary: We determine the centers of quasi-homogeneous polynomial planar vector fields of degree \(0, 1, 2, 3\) and \(4\). In addition, in every case we make a study of the reversibility and the analytical integrability of each one of the above centers. We find polynomial centers which are neither orbitally reversible nor analytically integrable, this is a new scenario in respect to the one of non-degenerate and nilpotent centers.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
37C99 Smooth dynamical systems: general theory
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