Hopf algebra techniques to handle dynamical systems and numerical integrators

TL;DR

This paper extends algebraic techniques based on Hopf algebras to analyze dynamical systems and numerical integrators, enabling more efficient computations through generalized algebraic frameworks.

Contribution

It introduces the extension of Hopf algebra techniques to more general structures, improving computational efficiency in dynamical systems analysis.

Findings

Extended techniques to general Hopf algebras.

Achieved more efficient computational methods.

Unified algebraic approach to dynamical systems.

Abstract

In a series of papers the present authors and their coworkers have developed a family of algebraic techniques to solve a number of problems in the theory of discrete or continuous dynamical systems and to analyze numerical integrators. Given a specific problem, those techniques construct an abstract, {\em universal} version of it which is solved algebraically; then, the results are tranferred to the original problem with the help of a suitable morphism. In earlier contributions, the abstract problem is formulated either in the dual of the shuffle Hopf algebra or in the dual of the Connes-Kreimer Hopf algebra. In the present contribution we extend these techniques to more general Hopf algebras, which in some cases lead to more efficient computations.