TY - JOUR
T1 - A note on the first integrals of the ABC system
AU - Llibre, Jaume
AU - Valls, Clàudia
PY - 2012/2/2
Y1 - 2012/2/2
N2 - 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.
AB - 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.
UR - https://ddd.uab.cat/record/150560
U2 - https://doi.org/10.1063/1.3682692
DO - https://doi.org/10.1063/1.3682692
M3 - Article
VL - 53
JO - Journal of Mathematical Physics
JF - Journal of Mathematical Physics
SN - 0022-2488
IS - 2
M1 - 023505
ER -