Limit cycles bifurcated from a reversible quadratic center. (English) Zbl 1142.34019

The main result of the paper is that under quadratic perturbations the exact upper bound of the number of limit cycles produced by the period annulus of the system
\[ \dot{z}=-iz(1+A\bar{z}) \]
with a nonzero complex number \(A\) is two. It follows basically from a careful estimate of the number of zeros of the associate Abelian integrals. Since there is no Picard-Fuchs equation in such case, the author uses different technique for this purpose.


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