Applicable formulae for the parameters μ2, β2, τ2, μ4, β4 and τ4 of N-dimensional Hopf bifurcation theory are presented. The center manifold theorem is used to reduce the system from N dimensions to 2 dimensions. Approximate solution of the system in Poincaré normal form provides the formulae. The formulae are explicit so that the parameters may be computed directly from partial derivatives of the system in real canonical form. The formula for μ2 is shown to be identical to that of I. D. Hsu and N. D. Kazarinoff. Formulae for the parameters μ1, μ2, μ3, τ1, τ2, τ3, β2, β3 and β4 in the "tangency" case Re λ1′(vc) = 0, Re λ1′'(vc) ≠ 0 are also presented. © 1978.
Hassard, B., & Wan, Y. H. (1978). Bifurcation formulae derived from center manifold theory. Journal of Mathematical Analysis and Applications, 63(1), 297–312. https://doi.org/10.1016/0022-247X(78)90120-8