Nondegeneracy (to determine criticality) requires the first Lyapunov coefficient 1\ell_11 (computed from second and third derivatives) to be nonzero. If 1<0\ell_1<01<0 the Hopf is supercritical (stable small amplitude limit cycles born); if 1>0\ell_1>01>0 it is subcritical (unstable cycles and often hysteresis).
We give the standard formula for 1\ell_11 below.
5. Formulae for normal-form coefficients (fold a,ba,ba,b and Hopf 1\ell_11)
(a) Saddle-node coefficients a,ba,ba,b
Right nullvector vvv and left nullvector www normalized so wv=1w^\top v = 1wv=1.
Quadratic form DX2F(Xc)[u,v]D_X^2 F(X_c)[u,v]DX2F(Xc)[u,v] is defined componentwise:
(DX2F(Xc)[u,v])i=j,k2Fixjxk(Xc)ujvk.\big(D_X^2 F(X_c)[u,v]\big)_i = \sum_{j,k} \frac{\partial^2 F_i}{\partial x_j \partial x_k}(X_c)\,u_j v_k.(DX2F(Xc)[u,v])i=j,kxjxk2Fi(Xc)ujvk.
Then
a=12wDX2F(Xc)[v,v],b=wF(Xc),a = \tfrac12\, w^\top D_X^2 F(X_c)[v,v],\qquad b = w^\top F_{\mu}(X_c),a=21wDX2F(Xc)[v,v],b=wF(Xc),
with F=F/EF_{\mu} = \partial F/\partial \mu_EF=F/E.
(These are exactly the coefficients that appear in the reduced scalar normal form y=ay2+b(c)+ \dot y = a y^2 + b (\mu-\mu_c) + \cdotsy=ay2+b(c)+.)
(b) Hopf first Lyapunov coefficient 1\ell_11 (Kuznetsov formula)
Let J=DXF(Xc)J = D_X F(X_c)J=DXF(Xc) have a simple pair of pure imaginary eigenvalues i0\pm i\omega_0i0 with right eigenvector vvv and left eigenvector www normalized as w,v=wv=1\langle w, v\rangle = w^\dagger v = 1w,v=wv=1. Define the multilinear forms BBB and CCC:
B(u,v)=DX2F(Xc)[u,v]B(u,v) \;=\; D_X^2 F(X_c)[u,v]B(u,v)=DX2F(Xc)[u,v] (vector valued bilinear form),
C(u,v,w)=DX3F(Xc)[u,v,w]C(u,v,w) \;=\; D_X^3 F(X_c)[u,v,w]C(u,v,w)=DX3F(Xc)[u,v,w] (vector valued trilinear form).
Then (see Kuznetsov, Elements of Applied Bifurcation Theory, 3.4) the first Lyapunov coefficient is
1=120{w,C(v,v,v)2w,B(v,(J2i0I)1B(v,v))+w,B(v,(J)1B(v,v))}.\boxed{% \ell_1 \;=\; \frac{1}{2\omega_0}\,\Re\Big\{\,\langle w,\, C(v,v,\bar v)\rangle - 2\langle w,\, B\big(v,\, (J-2i\omega_0 I)^{-1} B(v,v)\big)\rangle + \langle w,\, B\big(\bar v,\, (J)^{-1} B(v,\bar v)\big)\rangle \Big\}.}1=201{w,C(v,v,v)2w,B(v,(J2i0I)1B(v,v))+w,B(v,(J)1B(v,v))}.
Here:
v\bar vv is complex conjugate of vvv.
Inverse operators (J2i0I)1(J-2i\omega_0 I)^{-1}(J2i0I)1 and J1J^{-1}J1 act on the right-hand side vectors; these inverses exist under the generic Hopf nonresonance assumptions.
If 1<0\ell_1<01<0 the Hopf is supercritical.
This formula can be expanded componentwise using the explicit second and third derivatives of FFF. For a 3-D system it is straightforward but algebraically lengthy --- using symbolic algebra packages or finite-difference multilinear approximations is common in practice.
