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A centre theorem for two-dimensional complex holomorphic systems and its generalization. (English) Zbl 0835.34007
Summary: We consider a two-dimensional complex holomorphic system. In particular, we use the centre manifold theory together with the singular point theory of C. H. Briot and J. C. Bouquet [J. Éc. imp. Polyt. 21, 133 (1856)] to establish a centre theorem concerning the behaviour of the phase paths of the system in the neighbourhood of an equilibrium point having a single purely imaginary eigenvalue. An extended centre theorem is established for the corresponding \(N\)-dimensional complex holomorphic system \((N \geq 3)\).

34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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