Below we derive the normal forms for the two generic bifurcations we care about (saddle-node / fold and Hopf) for our unrest system. I keep the derivation general (valid for the 3--4D system introduced earlier) but give an explicit workflow and the exact algebraic expressions you must compute to obtain the reduced normal forms. That makes the derivation verifiable and ready to be implemented symbolically (e.g., in Mathematica/Maple/SymPy) or numerically.
Notation & setup
State vector XRnX\in\mathbb{R}^nXRn with n=3n=3n=3 or 444 (we typically use the reduced core X=(T,E,P)X=(T,E,P)^\topX=(T,E,P) or full X=(T,E,P,H)X=(T,E,P,H)^\topX=(T,E,P,H)).
Deterministic vector field X=F(X;,) \dot X = F(X;\mu,\Theta)X=F(X;,), where \mu is the primary scalar bifurcation parameter (here =E\mu=\mu_E=E) and \Theta denotes the leadership parameter vector.
Fixed point X(,)X^*(\mu,\Theta)X(,) satisfies F(X;,)=0F(X^*;\mu,\Theta)=0F(X;,)=0.
Jacobian J(X;,)=DXFXJ(X^*;\mu,\Theta) = D_XF|_{X^*}J(X;,)=DXFX. Eigenvalues i()\lambda_i(\mu)i() of JJJ determine linear stability.
We will derive the normal form in two cases:
a. Saddle-node (fold): a simple real eigenvalue passes through zero. The reduced normal form near the bifurcation is x=r+x2+...\dot x = r + \alpha x^2 + \ldotsx=r+x2+... (one-dimensional).
b. Hopf: a complex conjugate pair crosses the imaginary axis at i0\pm i\omega_0i0. The reduced normal form on the center manifold is z=(+i0)z+1zz2+...\dot z = (\eta + i\omega_0)z + \ell_1 z|z|^2 + \ldotsz=(+i0)z+1zz2+... where the real part of 1\ell_11 (the first Lyapunov coefficient) determines super/subcritical nature.
a. Saddle-node (fold) --- reduction & normal form
1. Existence / linear condition
Assume at =c\mu=\mu_c=c there is an equilibrium X=XcX^*=X_cX=Xc such that:
F(Xc;c)=0F(X_c;\mu_c)=0F(Xc;c)=0.
Jc:=J(Xc;c)J_c := J(X_c;\mu_c)Jc:=J(Xc;c) has a single simple zero eigenvalue, and all other eigenvalues have nonzero real parts. Let vvv be a right nullvector and www the corresponding left nullvector (row vector) normalized so that wv=1w^\top v = 1wv=1.
2. Nondegeneracy & transversality conditions
Two standard nondegeneracy conditions must hold for a classical saddle-node:
Nondegeneracy (quadratic nonlinearity):
a:=12wD2F(Xc)[v,v]0.a := \tfrac{1}{2}\, w^\top D^2F(X_c)[v,v] \;\neq\; 0.a:=21wD2F(Xc)[v,v]=0.
Here D2F(Xc)[v,v]D^2F(X_c)[v,v]D2F(Xc)[v,v] is the vector whose iii-th component is j,k2Fixjxk(Xc)vjvk\sum_{j,k}\frac{\partial^2 F_i}{\partial x_j\partial x_k}(X_c)\,v_j v_kj,kxjxk2Fi(Xc)vjvk.
Transversality (parameter dependence pushes eigenvalue through zero):
b:=wF(Xc;c)0.b := w^\top \partial_\mu F(X_c;\mu_c) \;\neq\; 0.b:=wF(Xc;c)=0.
If a0a\neq0a=0 and b0b\neq0b=0, the local dynamics on the one-dimensional center manifold reduce (after smooth coordinate change and parameter re-scaling) to:
y=b(c)+ay2+higher order terms.\dot y \;=\; b(\mu-\mu_c) + a\, y^2 + \text{higher order terms}.y=b(c)+ay2+higher order terms.
