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

TL;DR

This paper provides a comprehensive analysis of isochronous center families in holomorphic dynamical systems, utilizing Briot-Bouquet systems to establish existence results in arbitrary dimensions.

Contribution

It introduces a detailed study of Briot-Bouquet systems and applies this to prove the existence of isochronous centers in multi-dimensional holomorphic systems.

Findings

Complete solution for isochronous center families in holomorphic systems

Extension of Briot-Bouquet theory to arbitrary dimensions

Application to three-dimensional systems and beyond

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.