Integrability, degenerate centers, and limit cycles for a class of polynomial differential systems

J. Giné, J. Llibre

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9 Citations (Scopus)

Abstract

We consider the class of polynomial differential equations x ̇ Pn(x,y)+Pn+1(x,y)+Pn+2(x,y), y ̇=Qn(x,y)+Qn+1(x,y)+Qn+2(x,y), for n ≥ 1 and where Pi and Qi are homogeneous polynomials of degree i These systems have a linearly zero singular point at the origin if n > 2. Inside this class, we identify a new subclass of Darboux integrable systems, and some of them having a degenerate center, i.e., a center with linear part identically zero. Moreover, under additional conditions such Darboux integrable systems can have at most one limit cycle. We provide the explicit expression of this limit cycle. © 2006 Elsevier Ltd.
Original languageEnglish
Pages (from-to)1453-1462
JournalComputers and Mathematics with Applications
Volume51
Issue number9-10
DOIs
Publication statusPublished - 1 May 2006

Keywords

  • Algebraic limit cycle
  • Degenerate center
  • Integrability
  • Linearly zero singular point
  • Polynomial differential system
  • Polynomial vector field

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