Bifurcation of critical periods from the rigid quadratic isochronous vector field

A. Gasull, Y. Zhao

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22 Citations (Scopus)

Abstract

This paper is concerned with the study of the number of critical periods of perturbed isochronous centers. More concretely, if X0 is a vector field having an isochronous center of period T0 at the point p and Xε{lunate} is an analytic perturbation of X0 such that the point p is a center for Xε{lunate} then, for a suitable parameterization ξ of the periodic orbits surrounding p, their periods can be written as T (ξ, ε{lunate}) = T0 + T1 (ξ) ε{lunate} + T2 (ξ) ε{lunate}2 + ⋯. Firstly we give formulas for the first functions Tl (ξ) that can be used for quite general vector fields. Afterwards we apply them to study how many critical periods appear when we perturb the rigid quadratic isochronous center over(x, ̇) = - y + x y, over(y, ̇) = x + y2 inside the class of centers of the quadratic systems or of polynomial vector fields of a fixed degree. © 2007 Elsevier Masson SAS. All rights reserved.
Original languageEnglish
Pages (from-to)292-312
JournalBulletin des Sciences Mathematiques
Volume132
DOIs
Publication statusPublished - 1 Jun 2008

Keywords

  • Bifurcations
  • Critical periods
  • Isochronous center
  • Period function

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