Center problem for Λ–Ω differential systems

Jaume Llibre, Rafael Ramírez, Valentín Ramírez

Research output: Contribution to journalArticleResearch

1 Citation (Scopus)

Abstract

© 2019 Elsevier Inc. The Λ-Ω systems are the real planar polynomial differential equations of degree m x˙=−y(1+Λ)+xΩ,y˙=x(1+Λ)+yΩ, where Λ=Λ(x,y) and Ω=Ω(x,y) are polynomials of degree at most m−1 such that Λ(0,0)=Ω(0,0)=0. We study the center problem for these Λ-Ω systems. Any planar vector field with linear type center can be written as a Λ-Ω system if and only if the Poincaré-Liapunov first integral is of the form F= [Formula presented] (x2+y2)(1+O(x,y)). These kind of linear type centers are called weak centers, they contain the class of centers studied by Alwash and Lloyd [1], and also contain the uniform isochronous centers, and the holomorphic isochronous centers, but they do not coincide with the all class of isochronous centers. The main objective of this paper is to study the center problem for two particular classes of Λ-Ω systems of degree m. First if Λ=μ(a2x−a1y), and Ω=a1x+a2y+Ωm−1, where μ,a1,a2 are constants and Ωm−1=Ωm−1(x,y) is a homogenous polynomial of degree m−1, then we prove the following results. (i) These Λ-Ω systems have a weak center at the origin if and only if (μ+m−2)(a12+a22)=0, and ∫02πΩm−1(cos⁡t,sin⁡t)dt=0; (ii) If m=2,3,4,5,6 and (μ+m−2)(a12+a22)≠0, then the given Λ–Ω systems have a weak center at the origin if and only if these systems after a linear change of variables (x,y)⟶(X,Y) are invariant under the transformations (X,Y,t)⟶(−X,Y,−t). Second if Λ=a1x+a2y, and Ω=Ωm−1, where a1,a2 are constants and Ωm−1=Ωm−1(x,y) is a homogenous polynomial of degree m−1, then we prove the following results. (i) These Λ-Ω systems have a weak center at the origin if and only if a1=a2=0, and ∫02πΩm−1(cos⁡t,sin⁡t)dt=0; (ii) If m=2,3,4,5 and a12+a22≠0, then the given Λ–Ω systems have a weak center at the origin if and only if these systems after a linear change of variables (x,y)⟶(X,Y) are invariant under the transformations (X,Y,t)⟶(−X,Y,−t). We observe that the main difficulty to prove results (ii) for m>6 is related with the huge computations necessary for proving them.
Original languageEnglish
Pages (from-to)6409-6446
JournalJournal of Differential Equations
Volume267
DOIs
Publication statusPublished - 15 Nov 2019

Keywords

  • Analytic planar differential system
  • Linear type center
  • Reversible system
  • Weak center

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