Usual normal form scaling yields the canonical fold:
x=r+x2,rb(c),xy.\dot x = r + x^2,\qquad r \propto b(\mu-\mu_c),\;x\propto y.x=r+x2,rb(c),xy.
Interpretation: the sign of aba bab tells whether equilibria are created/destroyed for \mu increasing past c\mu_cc. For our problem, computing aaa and bbb (explicitly in terms of partial derivatives) gives the analytic expression for crit()\mu_{\mathrm{crit}}(\Theta)crit() locally.
3. How to compute aaa and bbb concretely
Solve Jcv=0J_c v = 0Jcv=0 for right nullvector vvv.
Solve wJc=0w^\top J_c = 0wJc=0 for left nullvector www, normalize wv=1w^\top v = 1wv=1.
Compute the vector D2F(Xc)[v,v]D^2F(X_c)[v,v]D2F(Xc)[v,v] with components
(D2F(Xc)[v,v])i=j,k=1n2Fixjxk(Xc)vjvk.\big(D^2F(X_c)[v,v]\big)_i \;=\; \sum_{j,k=1}^n \frac{\partial^2 F_i}{\partial x_j\partial x_k}(X_c)\, v_j v_k.(D2F(Xc)[v,v])i=j,k=1nxjxk2Fi(Xc)vjvk.
Then a=12wD2F(Xc)[v,v]a = \tfrac{1}{2}\, w^\top D^2F(X_c)[v,v]a=21wD2F(Xc)[v,v].
Compute F\partial_\mu FF (partial derivative of vector field w.r.t. \mu); evaluate at (Xc,c)(X_c,\mu_c)(Xc,c) and form b=wF(Xc;c)b = w^\top \partial_\mu F(X_c;\mu_c)b=wF(Xc;c).
If aaa and bbb satisfy the inequalities above, you have the fold normal form.
Practical note: For our model the only explicit dependence on \mu is in E\dot EE via the additive E\mu_EE term, so F=eE\partial_\mu F = e_EF=eE unit vector in EEE-direction; thus b=wEb = w_Eb=wE (the EEE-component of the left nullvector), making the transversality test computationally straightforward.
b. Hopf bifurcation --- reduction & normal form
1. Linear condition
Assume at =h\mu=\mu_h=h there is equilibrium X=XhX^*=X_hX=Xh such that:
F(Xh;h)=0F(X_h;\mu_h)=0F(Xh;h)=0.
Jacobian JhJ_hJh has a simple pair of purely imaginary eigenvalues 1,2(h)=i0\lambda_{1,2}(\mu_h)=\pm i\omega_01,2(h)=i0 with 0>0\omega_0>00>0; all other eigenvalues have nonzero real parts. The pair is simple (algebraic multiplicity 1).
2. Transversality & nondegeneracy
