TY - JOUR
T1 - Darboux integrability and algebraic limit cycles for a class of polynomial differential systems
AU - Cao, Jin Long
AU - Llibre, Jaume
AU - Zhang, Xiang
PY - 2014/1/1
Y1 - 2014/1/1
N2 - This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems ẋ = λ x-y+P n+1(x, y)+xF 2n(x, y), ẏ = x+λy+Q n+1(x, y)+yF 2n(x, y), where P i(x, y), Q i(x, y) and F i(x, y) are homogeneous polynomials of degree i. Within this class, we identify some new Darboux integrable systems having either a focus or a center at the origin. For such Darboux integrable systems having degrees 5 and 9 we give the explicit expressions of their algebraic limit cycles. For the systems having degrees 3, 5, 7 and 9 and restricted to a certain subclass we present necessary and sufficient conditions for being Darboux integrable. © 2014 Science China Press and Springer-Verlag Berlin Heidelberg.
AB - This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems ẋ = λ x-y+P n+1(x, y)+xF 2n(x, y), ẏ = x+λy+Q n+1(x, y)+yF 2n(x, y), where P i(x, y), Q i(x, y) and F i(x, y) are homogeneous polynomials of degree i. Within this class, we identify some new Darboux integrable systems having either a focus or a center at the origin. For such Darboux integrable systems having degrees 5 and 9 we give the explicit expressions of their algebraic limit cycles. For the systems having degrees 3, 5, 7 and 9 and restricted to a certain subclass we present necessary and sufficient conditions for being Darboux integrable. © 2014 Science China Press and Springer-Verlag Berlin Heidelberg.
KW - Abel differential equation
KW - Darboux first integral
KW - algebraic limit cycles
UR - https://ddd.uab.cat/record/150749
U2 - https://doi.org/10.1007/s11425-014-4772-8
DO - https://doi.org/10.1007/s11425-014-4772-8
M3 - Article
VL - 57
SP - 775
EP - 794
JO - Science China Mathematics
JF - Science China Mathematics
SN - 1674-7283
IS - 4
ER -