Linearised Higher Variational Equations

TL;DR

This paper develops explicit, computationally friendly forms for linearised higher-order variational equations and their monodromy matrices in autonomous systems, aiding the analysis of integrability via Ziglin-Morales-Ramis theory.

Contribution

It introduces a compact, explicit formulation of higher variational equations and their monodromy matrices, facilitating their computation and analysis.

Findings

Provides a new explicit form for variational systems

Simplifies systems for Taylor coefficients of first integrals

Enhances tools for studying dynamical system integrability

Abstract

This work explores the tensor and combinatorial constructs underlying the linearised higher-order variational equations of a generic autonomous system along a particular solution. The main result of this paper is a compact yet explicit and computationally amenable form for said variational systems and their monodromy matrices. Alternatively, the same methods are useful to retrieve, and sometimes simplify, systems satisfied by the coefficients of the Taylor expansion of a formal first integral for a given dynamical system. This is done in preparation for further results within Ziglin-Morales-Ramis theory, specifically those of a constructive nature.