A CENTER THEOREM FOR 2-DIMENSIONAL COMPLEX HOLOMORPHIC SYSTEMS AND ITS GENERALIZATION

We consider a two-dimensional complex holomorphic system. In particula r, we use the centre manifold theory together with the singular point theory of C. H. Briot & J. C. Bouquet (J. Ec. imp. Polyt. 21. 133 (185 6)) to establish a centre theorem concerning the behaviour of the phas e paths of the system in the neighbourhood of an equilibrium point hav ing a single purely imaginary eigenvalue. An extended centre theorem i s established for the corresponding N-dimensional complex holomorphic system (N greater than or equal to 3).