The Briot–Bouquet systems and the center families for holomorphic dynamical systems

Abstract

We give a complete solution to the existence of isochronous center families for holomorphic dynamical systems. The study of center families for n-dimensional holomorphic dynamical systems naturally leads to the study of (n−1)-dimensional Briot–Bouquet systems in the phase space. We first give a detailed study of the Briot–Bouquet systems. Then we show the existence of isochronous center families in the neighborhood of the equilibrium point of three-dimensional systems based on the two-dimensional Briot–Bouquet theory. The same approach works in arbitrary dimensions.