corresponding Magnus expansion; and of their relations, into the unified algebraic setting of Rota–Baxter maps and their inverse skewderivations. This picture clarifies the approximate solutions to generalized integral equations corresponding to non-autonomous linear (skew)differential equations.'
author:
- |
    José F. Cariñena\*, Kurusch Ebrahimi-Fard, \
    Héctor Figueroa and José M. Gracia-Bondía§$\|$\
    \*Departamento de Física Teórica, Universidad de Zaragoza,\
    Zaragoza 50006, Spain\
    Max-Planck-Institut für Mathematik, Vivatsgasse 7\
    D-53111 Bonn, Germany\
    Departamento de Matemáticas, Universidad de Costa Rica,\
    San Pedro 2060, Costa Rica\
    §Departamento de Física Teórica I, Universidad Complutense,\
    Madrid 28040, Spain\
    $\|$Departamento de Física, Universidad de Costa Rica,\
    San Pedro 2060, Costa Rica
date: 'December 28, 2006 '
title: Hopf algebras in dynamical systems theory
---

*Mathematics Subject Classification 2000*: 16W25, 16W30, 37B55, 37C10

*2006 PACS*: 02.20.Qs, 02.30.Hq, 02.40.Gh, 45.30.+s

Keywords: Differential equations, Lie–Scheffers systems, Rota–Baxter operators, Hopf algebra, Spitzer’s identity, Magnus expansion

Aim, plan of the article and preliminaries
==========================================

This paper studies non-autonomous differential equations of the general type $$\begin{aligned}
\dot g(t)g^{-1}(t) &= \xi(t),  \qquad g(t_0) = 1_G {\quad\mbox{or}\quad}
\label{eq:gato-encerrado}
\\
g^{-1}(t)\dot g(t) &= \eta(t), \qquad g(t_0) = 1_G,
\label{eq:menos-gatos}\end{aligned}$$ where the unknown $g:{\mathbb{R}}_t\to G$ is a curve on a (maybe infinite-dimensional) local Lie group $G$, with $1_G$ the 