TY - JOUR

T1 - Limit cycles of polynomial differential systems bifurcating from the periodic orbits of a linear differential system in Rd

AU - Llibre, Jaume

AU - Makhlouf, Amar

PY - 2009/9/1

Y1 - 2009/9/1

N2 - 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.

AB - 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.

KW - Averaging method

KW - Bifurcation

KW - Limit cycles

KW - Polynomial vector fields

U2 - https://doi.org/10.1016/j.bulsci.2009.04.004

DO - https://doi.org/10.1016/j.bulsci.2009.04.004

M3 - Article

VL - 133

SP - 578

EP - 587

JO - Bulletin des Sciences Mathematiques

JF - Bulletin des Sciences Mathematiques

SN - 0007-4497

ER -