TY - JOUR
T1 - On a conjecture on the integrability of Liénard systems
AU - Llibre, Jaume
AU - Murza, Adrian C.
AU - Valls, Claudia
N1 - Publisher Copyright:
© 2019, Springer-Verlag Italia S.r.l., part of Springer Nature.
PY - 2020/4/1
Y1 - 2020/4/1
N2 - We consider the Liénard differential systems [Equation not available: see fulltext.]in C2 where F(x) is an analytic function satisfying F(0) = 0 and F′(0) ≠ 0. Then these systems have a strong saddle at the origin of coordinates. It has been conjecture that if such systems have an analytic first integral defined in a neighborhood of the origin, then the function F(x) is linear, i.e. F(x) = ax. Here we prove this conjecture, and show that when F(x) is linear and system (1) has an analytic first integral, this is a polynomial.
AB - We consider the Liénard differential systems [Equation not available: see fulltext.]in C2 where F(x) is an analytic function satisfying F(0) = 0 and F′(0) ≠ 0. Then these systems have a strong saddle at the origin of coordinates. It has been conjecture that if such systems have an analytic first integral defined in a neighborhood of the origin, then the function F(x) is linear, i.e. F(x) = ax. Here we prove this conjecture, and show that when F(x) is linear and system (1) has an analytic first integral, this is a polynomial.
KW - First integral
KW - Liénard system
KW - Strong saddle
UR - https://www.scopus.com/pages/publications/85081019044
U2 - 10.1007/s12215-018-00398-6
DO - 10.1007/s12215-018-00398-6
M3 - Article
AN - SCOPUS:85081019044
SN - 0009-725X
VL - 69
SP - 209
EP - 216
JO - Rendiconti del Circolo Matematico di Palermo
JF - Rendiconti del Circolo Matematico di Palermo
IS - 1
ER -