Hopf bifurcation for degenerate singular points of multiplicity 2n−12n−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 2n−12n−1 in dimension 3 via averaging theory. More specifically, we consider the systemx˙=−Hy(x,y)+P2n(x,y,z)+εP2n−1(x,y),y˙=Hx(x,y)+Q2n(x,y,z)+εQ2n−1(x,y),z˙=R2n(x,y,z)+εcz2n−1, whereH=12n(x2l+y2l)m,n=lm,P2n−1=x(p1x2n−2+p2x2n−3y+⋯+p2n−1y2n−2),Q2n−1=y(p1x2n−2+p2x2n−3y+⋯+p2n−1y2n−2), and P2nP2n, Q2nQ2n and R2nR2n 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=0ε=0 to ε≠0ε≠0 sufficiently small, from the origin it can bifurcate 2n−12n−1 limit cycles, and that using the averaging theory of second order from the origin it can bifurcate 3n−13n−1 limit cycles when l=1l=1.