Formal first integrals and higher variational equations

TL;DR

This paper develops a formal approach using differential Galois theory and higher variational equations to find first integrals in dynamical systems, applicable beyond Hamiltonian cases, with explicit examples.

Contribution

It introduces a unified linear system framework for higher variational equations that captures all first integrals, extending applicability beyond Hamiltonian systems.

Findings

Explicit first integrals for Dixon’s system and SIR model

Unified linear system for higher variational equations

Application to non-Hamiltonian systems like Van der Pol oscillator

Abstract

The question of how Algebra can be used to solve dynamical systems and characterize chaos was first posed in a fertile mathematical context by Ziglin, Morales, Ramis and Sim\'o using differential Galois theory. Their study was aimed at first-order, later higher-order, variational equations of Hamiltonian systems. Recent work by this author formalized a compact yet comprehensive expression of higher-order variationals as one infinite linear system, thereby simplifying the approach. More importantly, the dual of this linear system contains all information relevant to first integrals, regardless of whether the original system is Hamiltonian. This applicability to formal calculation of conserved quantities is the centerpiece of this paper, following an introduction to the requisite context. Three important examples, namely particular cases of Dixon's system, the SIR epidemiological model with vital dynamics and the Van der Pol oscillator, are tackled, and explicit convergent first integrals are provided for the first two.