The isochronous center on center manifolds for three dimensional differential systems

TL;DR

This paper introduces a direct method to identify isochronous centers on center manifolds in three-dimensional polynomial differential systems, simplifying the process without computing the full center manifold.

Contribution

It develops a new approach using isochronous constants and recursive formulas, extending the formal series method for three-dimensional systems and enabling computer algebra implementation.

Findings

Defined isochronous constants and derived recursive formulas.

Provided conditions for isochronous centers without computing center manifolds.

Applied the method to specific systems demonstrating its effectiveness.

Abstract

In this paper, we give a direct method to study the isochronous centers on center manifolds of three dimensional polynomial differential systems. Firstly, the isochronous constants of the three dimensional system are defined and its recursive formulas are given. The conditions of the isochronous center are determined by the computation of isochronous constants in which it doesn't need compute center manifolds of three dimensional systems. Then the isochronous center conditions of two specific systems are discussed as the applications of our method. The method is an extension and development of the formal series method for the fine focus of planar differential systems and also readily done with using computer algebra system such as Mathematica or Maple.