A note on the first integrals of the ABC system

Jaume Llibre, Clàudia Valls

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3 Citations (Scopus)

Abstract

Without loss of generality the ABC systems reduce to two cases: either A = 0 and B, C ≥ 0, or A = 1 and 0 < B, C ≤ 1. In the first case it is known that the ABC system is completely integrable, here we provide its explicit first integrals. In the second case Ziglin ["Dichotomy of the separatrices and the nonexistence of first integrals in systems of differential equations of Hamiltonian type with two degrees of freedom," Izv. Akad. Nauk SSSR, Ser. Mat.51, 1088 (1987)] proved that the ABC system with 0 < B < 1 and C > 0 sufficiently small has no real meromorphic first integrals. We improve Ziglin's result showing that there are no C 1 first integrals under convenient assumptions. © 2012 American Institute of Physics.
Original languageEnglish
Article number023505
JournalJournal of Mathematical Physics
Volume53
Issue number2
DOIs
Publication statusPublished - 2 Feb 2012

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