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 -