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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 \((N \geq 3)\).

MSC:
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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