Hopf bifurcation for degenerate singular points of multiplicity 2 n - 1 in dimension 3

The main purpose of this paper is to study the Hopf bifurcation for a class of degenerate singular points of multiplicity 2 n - 1 in dimension 3 via averaging theory. More specifically, we consider the systemover(x, ̇) = - Hy (x, y) + P2 n (x, y, z) + ε P2 n - 1 (x, y), over(y, ̇) = Hx (x, y) + Q2 n (x, y, z) + ε Q2 n - 1 (x, y), over(z, ̇) = R2 n (x, y, z) + ε c z2 n - 1, whereH = frac(1, 2 n) (x2 l + y2 l)m, n = l m, P2 n - 1 = x (p1 x2 n - 2 + p2 x2 n - 3 y + ⋯ + p2 n - 1 y2 n - 2), Q2 n - 1 = y (p1 x2 n - 2 + p2 x2 n - 3 y + ⋯ + p2 n - 1 y2 n - 2), and P2 n, Q2 n and R2 n are arbitrary analytic functions starting with terms of degree 2n. We prove using the averaging theory of first order that, moving the parameter ε from ε = 0 to ε ≠ 0 sufficiently small, from the origin it can bifurcate 2 n - 1 limit cycles, and that using the averaging theory of second order from the origin it can bifurcate 3 n - 1 limit cycles when l = 1. © 2007 Elsevier Masson SAS. All rights reserved.