Shuffle and Fa\`a di Bruno Hopf Algebras in the Center Problem for Ordinary Differential Equations

TL;DR

This paper explores the Hopf algebra framework for analyzing the center problem in certain differential equations, extending previous methods and examining the combinatorial structure of the first return map.

Contribution

It introduces a Hopf algebra approach to the center problem and extends existing combinatorial methods for analyzing the first return map.

Findings

Hopf algebra approach effectively describes the center problem

Extended combinatorial analysis of the first return map

Unified framework for previous methods

Abstract

In this paper we describe the Hopf algebra approach to the center problem for the differential equation $\frac{dv}{dx}=\sum_{i=1}^{\infty}a_{i}(x)v^{i+1}$, $x\in [0,T]$, and study some combinatorial properties of the first return map of this equation. The paper summarizes and extends previously developed approaches to the center problem due to Devlin and the author.