Formal first integrals and higher variational equations

Sergi Simon; Sergi Simon

doi:10.3934/era.2026358

Electronic Research Archive

2026, Volume 34, Issue 11: 8455-8487. doi: 10.3934/era.2026358

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Research article

Formal first integrals and higher variational equations

Sergi Simon ^,

School of Computing, Mathematics & Physics, Faculty of Technology, University of Portsmouth, Lion Gate Building, Lion Terrace, Portsmouth PO1 3HF, UK

Received: 04 June 2026 Revised: 27 August 2026 Accepted: 24 September 2026 Published: 30 September 2026

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 Simó using differential Galois theory. Their study was aimed at first-order, and 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 formal 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 susceptible-infected-recovered (SIR) epidemiological model with vital dynamics, and the Van der Pol oscillator, were tackled, and explicit convergent first integrals were provided for the first two.
- integrability,
- first integrals,
- formal calculus,
- chaos,
- dynamical systems,
- Dixon's system,
- SIR epidemiological model,
- van der Pol oscillator
Citation: Sergi Simon. Formal first integrals and higher variational equations[J]. Electronic Research Archive, 2026, 34(11): 8455-8487. doi: 10.3934/era.2026358

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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 Simó using differential Galois theory. Their study was aimed at first-order, and 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 formal 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 susceptible-infected-recovered (SIR) epidemiological model with vital dynamics, and the Van der Pol oscillator, were tackled, and explicit convergent first integrals were provided for the first two.

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