Formal first integrals and higher variational equations

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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.

Keywords

integrability / first integrals / formal calculus / chaos / dynamical systems / Dixon’s system / SIR epidemiological model / van der Pol oscillator

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