TY - JOUR
T1 - Classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree
AU - Llibre, Jaume
AU - Valls, Clàudia
PY - 2011/5/1
Y1 - 2011/5/1
N2 - In this paper we classify the centers localized at the origin of coordinates, and their isochronicity for the polynomial differential systems in R2 of degree d that in complex notation z=x+iy can be written as. ż=(λ+i)z+Az(d-n+1)/2z(d+n-1)/2+Bz(d+n+1)/2z(d+n-1)/2+Cz(d+1)/2z(d-1)/2+Dz(d(2+j)n+1)/2z(d+(2+j)n-1)/2, where j is either 0 or 1. If j=0 then d≥5 is an odd integer and n is an even integer satisfying 2≤n≤(d+1)/2. If j=1 then d≥3 is an integer and n is an integer with converse parity with d and satisfying 0<n≥[(d+1)/3] where [.] denotes the integer part function. Furthermore λ∈R and A,B,C,D∈C. Note that if d=3 and j=0, we are obtaining the generalization of the polynomial differential systems with cubic homogeneous nonlinearities studied in K.E. Malkin (1964) [17], N.I. Vulpe and K.S. Sibirskii (1988) [25], J. Llibre and C. Valls (2009) [15], and if d=2, j=1 and C=0, we are also obtaining as a particular case the quadratic polynomial differential systems studied in N.N. Bautin (1952) [2], H. Zoladek (1994) [26]. So the class of polynomial differential systems here studied is very general having arbitrary degree and containing the two more relevant subclasses in the history of the center problem for polynomial differential equations. © 2011 Elsevier Inc.
AB - In this paper we classify the centers localized at the origin of coordinates, and their isochronicity for the polynomial differential systems in R2 of degree d that in complex notation z=x+iy can be written as. ż=(λ+i)z+Az(d-n+1)/2z(d+n-1)/2+Bz(d+n+1)/2z(d+n-1)/2+Cz(d+1)/2z(d-1)/2+Dz(d(2+j)n+1)/2z(d+(2+j)n-1)/2, where j is either 0 or 1. If j=0 then d≥5 is an odd integer and n is an even integer satisfying 2≤n≤(d+1)/2. If j=1 then d≥3 is an integer and n is an integer with converse parity with d and satisfying 0<n≥[(d+1)/3] where [.] denotes the integer part function. Furthermore λ∈R and A,B,C,D∈C. Note that if d=3 and j=0, we are obtaining the generalization of the polynomial differential systems with cubic homogeneous nonlinearities studied in K.E. Malkin (1964) [17], N.I. Vulpe and K.S. Sibirskii (1988) [25], J. Llibre and C. Valls (2009) [15], and if d=2, j=1 and C=0, we are also obtaining as a particular case the quadratic polynomial differential systems studied in N.N. Bautin (1952) [2], H. Zoladek (1994) [26]. So the class of polynomial differential systems here studied is very general having arbitrary degree and containing the two more relevant subclasses in the history of the center problem for polynomial differential equations. © 2011 Elsevier Inc.
KW - Centers of arbitrary degree
KW - Centers of polynomial vector fields
KW - Isochronous centers
U2 - https://doi.org/10.1016/j.aim.2011.02.003
DO - https://doi.org/10.1016/j.aim.2011.02.003
M3 - Article
VL - 227
SP - 472
EP - 493
JO - Advances in Mathematics
JF - Advances in Mathematics
SN - 0001-8708
ER -