Classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree

Jaume Llibre, Clàudia Valls

Research output: Contribution to journalArticleResearchpeer-review

11 Citations (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)472-493
JournalAdvances in Mathematics
Volume227
DOIs
Publication statusPublished - 1 May 2011

Keywords

  • Centers of arbitrary degree
  • Centers of polynomial vector fields
  • Isochronous centers

Fingerprint Dive into the research topics of 'Classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree'. Together they form a unique fingerprint.

Cite this