Bifurcation formulae derived from center manifold theory

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′(v_c) = 0, Re λ₁′'(v_c) ≠ 0 are also presented.