Limit cycles of polynomial differential systems bifurcating from the periodic orbits of a linear differential system in Rd

Jaume Llibre, Amar Makhlouf

Research output: Contribution to journalArticleResearchpeer-review

2 Citations (Scopus)

Abstract

Let Pk (x1, ..., xd) and Qk (x1, ..., xd) be polynomials of degree nk for k = 1, 2, ..., d. Consider the polynomial differential system in Rd defined byover(x, ̇)1 = - x2 + ε P1 (x1, ..., xd) + ε2 Q1 (x1, ..., xd),over(x, ̇)2 = x1 + ε P2 (x1, ..., xd) + ε2 Q2 (x1, ..., xd),over(x, ̇)k = ε Pk (x1, ..., xd) + ε2 Qk (x1, ..., xd), for k = 3, ..., d. Suppose that nk = n ≥ 2 for k = 1, 2, ..., d. Then, by applying the first order averaging method this system has at most (n - 1) nd - 2 / 2 limit cycles bifurcating from the periodic orbits of the same system with ε = 0; and by applying the second order averaging method it has at most (n - 1) (2 n - 1)d - 2 limit cycles bifurcating from the periodic orbits of the same system with ε = 0. We provide polynomial differential systems reaching these upper bounds. In fact our results are more general, they provide the number of limit cycles for arbitrary nk. © 2009 Elsevier Masson SAS. All rights reserved.
Original languageEnglish
Pages (from-to)578-587
JournalBulletin des Sciences Mathematiques
Volume133
DOIs
Publication statusPublished - 1 Sep 2009

Keywords

  • Averaging method
  • Bifurcation
  • Limit cycles
  • Polynomial vector fields

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