TY - JOUR
T1 - The 16th Hilbert problem restricted to circular algebraic limit cycles
AU - Llibre, Jaume
AU - Ramírez, Rafael
AU - Ramírez, Valentín
AU - Sadovskaia, Natalia
PY - 2016/4/5
Y1 - 2016/4/5
N2 - © 2015 Elsevier Inc. We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S-1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S-1 algebraic limit cycles given by circles, and this number is reached.
AB - © 2015 Elsevier Inc. We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S-1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S-1 algebraic limit cycles given by circles, and this number is reached.
KW - 16th Hilbert's problem
KW - Algebraic limit circles
KW - Darboux integrability
KW - Invariant algebraic circles
KW - Planar polynomial differential system
KW - Polynomial vector fields
U2 - 10.1016/j.jde.2015.12.019
DO - 10.1016/j.jde.2015.12.019
M3 - Article
SN - 0022-0396
VL - 260
SP - 5726
EP - 5760
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 7
ER -