# 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 language English 578-587 Bulletin des Sciences Mathematiques 133 https://doi.org/10.1016/j.bulsci.2009.04.004 Published - 1 Sep 2009

## Keywords

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

## Fingerprint

Dive into the research topics of 'Limit cycles of polynomial differential systems bifurcating from the periodic orbits of a linear differential system in R<sup>d</sup>'. Together they form a unique fingerprint.