Complete Folds and the Mixed Critical Coordinate

3. Stationary geometry and complete fold data

At fixed amplitude the memory Hessian and stationary solution are

$$(H_m)_{ee}=\kappa+\tfrac12e^{m_e}|b_e^Tz|^2>0.\tag{14}$$ $$m_e=-s_e,\quad s_ee^{s_e}=|b_e^Tz|^2/(2\kappa),\quad s_e=W_0(|b_e^Tz|^2/(2\kappa)).\tag{15}$$ $$V_{\rm eff}(z,a)=\tfrac\mu2\lVert z\rVert^2+ \sum_e\kappa(s_e+s_e^2/2)-\operatorname{Re}[(av_3/c)^*z].\tag{16}$$

$W_0$ is the principal real Lambert function (Corless et al., 1996). This is an exact stationary correspondence. For $w=e^{-s}$ an edge has tangential curvature $w$ and radial curvature $-Q_0(s)$:

$$Q_0(s)=e^{-s}\frac{s-1}{s+1},\quad Q_0'(s)=e^{-s}\frac{3-s^2}{(1+s)^2},\quad \chi=(2-\sqrt3)e^{-\sqrt3}=\max_{s\ge0}Q_0(s).\tag{17}$$

The orthonormal modes are $v_0=(1,1,1)^T/\sqrt3$, $v_1=(1,0,-1)^T/\sqrt2$ and $v_3$, with eigenvalues $0,1,3$.

Lemma 3.1 (Complete counted regime). For $\chi<\mu<3\chi$ and $a>0$, every stationary state has

$$z=\theta R(s)v_3,\qquad m=(-s,-s),\qquad \theta=|c|/c,\quad R(s)=\sqrt{\tfrac{4\kappa}{3}se^s}.\tag{18}$$ $$a(s)=|c|\sqrt{H_3(s)},\qquad H_3(s)=\tfrac{4\kappa}{3}se^s(\mu+3e^{-s})^2.\tag{19}$$

The two roots $s_-\in(1,\sqrt3)$ and $s_+>\sqrt3$ of $Q_0(s_\pm)=\mu/3$ give

$$a_{\rm up}=a(s_-),\qquad a_{\rm down}=a(s_+),\qquad a_{\rm down}<a_{\rm up}.\tag{20}$$

There are exactly two stable minima and an index-one saddle in the open interval. At either endpoint there is a simple saddle-node and the other minimum; outside the closed interval there is one stable equilibrium.

Proof. Write $z=tv_0+uv_3+hv_1$ in complex coordinates. The collective potential $\mu|t|^2/2$ forces $t=0$. Reflection fixes $u$ and reverses $h$. The even real two-dimensional $h$ slice has Hessian at least $(\mu-\chi)I>0$ since $Lv_1=v_1$. Its sole critical point is $h=0$. Both memory values then coincide; the amplitude equation fixes phase and gives (18)–(19). This excludes hidden complex or symmetry-broken equilibria. The derivative is

$$H_3'(s)=\tfrac{4\kappa}{3}(\mu+3e^{-s})e^s(1+s)[\mu-3Q_0(s)].\tag{21}$$

The unique maximum of $Q_0$ gives two simple turning points, maximum at $s_-$ and minimum at $s_+$. With $H_3(0)=0$ and $H_3(s)\to\infty$ the monotonic pieces give all counts. The complete Hessian is

$$H=\begin{pmatrix}K_R&B\\ B^T&H_m\end{pmatrix},\qquad K_R=\operatorname{diag}(K,K),\qquad B_{:e}=e^{m_e}\operatorname{diag}(B_e,B_e)q.\tag{22}$$

Completing the memory square gives the negative and zero counts of $S=K_R-BH_m^{-1}B^T=\nabla^2V_{\rm eff}$ and two additional positive directions. At an equal-edge state the six Schur eigenvalues are

$$\mu,\quad\mu-Q_0,\quad\mu-3Q_0;\qquad\mu,\quad\mu+w,\quad\mu+3w.\tag{23}$$

The full mobility bridge applies because $H$ is symmetric and $\operatorname{Sym}D>0$. A nonzero imaginary eigenvector would give $(Hv)^*D(Hv)=-i\omega v^*Hv$, impossible by real parts. A generalized zero vector yields $0=-v_0^*D^{-1}v_0$ for $Hv_0=0$, impossible because $D^{-1}$ has positive symmetric part. Deforming $D$ to $I$ therefore preserves half-plane counts and zero multiplicity, including complex algebraic multiplicities. Equation (23) gives all stability classes. At a fold only $\mu-3Q_0$ vanishes, leaving seven decaying modes. The nonzero saddle-node coefficients follow next. $\square$

At either fold,

$$X(x,a)=-DY(x,a),\quad Y(x,a)=\nabla F(x)-ah,\quad h=(\operatorname{Re}(v_3/c),\operatorname{Im}(v_3/c),0,0).\tag{24}$$

Let $x_f=x(s_f)$, $a_f=a(s_f)$ and $L_f=X_x(x_f,a_f)=-DH$. Normalize by

$$r=x_s(s_f),\qquad d_f=r^TD^{-1}r>0,\qquad l^T=r^TD^{-1}/d_f,\qquad l^Tr=1.\tag{25}$$

The two memory components of $r$ are $(-1,-1)$; neither amplitude quadrature nor memory is dropped. Differentiating the stationary identity at $a'=0$ gives $Hr=0$ and $r^TY_{xx}[r,r]=(r^Th)a''$; $r^Th=R'/|c|>0$. The full coefficients are

$$A_f=l^TX_a=\frac{r^Th}{d_f}>0,\qquad B_f=\tfrac12l^TX_{xx}[r,r]=-\tfrac12A_f a''(s_f).\tag{26}$$

From (21) $a''<0$ at $s_-$ and $a''>0$ at $s_+$. Thus $B_f$ is positive at the upper fold and negative at the lower fold. The simple zero and nonzero coefficients establish full saddle-nodes.

Table 1. Complete fold coefficients for (6) with normalization (25). $K_f=\zeta_1/(A_f|B_f|)^{1/3}$. The computed stable gap is 0.06.

4. Fixed-fold reduction with dynamic memory

4.1 Projection and extended nilpotent structure

Fix $P_f=rl^T$, $Q_f=I-P_f$. Since $L_fr=0$ and $l^TL_f=0$, $E_s=\ker l^T$ is invariant and its restricted $A_s$ is Hurwitz. All coordinates are fixed at the fold:

$$x-x_f=ry+w,\quad y=l^T(x-x_f),\quad w\in E_s,\quad \lambda=a-a_f.\tag{27}$$

Input is affine and state independent, so $X_{xa}=0$. Taylor expansion gives

$$\begin{aligned} \dot y&=A_f\lambda+B_fy^2+N_y(y,w),\quad \dot w=A_sw+Q_fX_a\lambda+N_w(y,w),\\ N_y&=O(|y|^3+|y|\lVert w\rVert+\lVert w\rVert^2),\\ N_w&=O(y^2+|y|\lVert w\rVert+\lVert w\rVert^2). \end{aligned}\tag{28}$$ $$\dot\lambda=\epsilon,\qquad \dot\epsilon=0.\tag{29}$$

The augmented constant-slope system has centre coordinates $(y,\lambda,\epsilon)$ with a nilpotent linear chain $\dot y=A_f\lambda$, $\dot\lambda=\epsilon$, rather than three independent semisimple zero modes. The finite-dimensional centre-manifold theorem applies to the zero-real-part centre spectrum and strictly stable complement (Carr, 1981, 2006). Choose a finite $C^4$ chart and graph

$$w=\psi(y,\lambda,\epsilon),\qquad \psi=O(y^2+|\lambda|+|\epsilon|).\tag{30}$$

Inserting $\psi=C_\lambda\lambda+C_\epsilon\epsilon+\cdots$ in the invariance equation gives

$$C_\lambda=\psi_\lambda(0)=-A_s^{-1}Q_fX_a,\qquad C_\epsilon=\psi_\epsilon(0)=A_s^{-1}C_\lambda.\tag{31}$$

The inverses act on $E_s$; $C_\epsilon$ retains parameter motion. The mixed critical equation becomes

$$\dot y=A_f\lambda+B_fy^2+R,\qquad |R|\le C(|y|^3+|y\lambda|+\lambda^2+|\epsilon|).\tag{32}$$

The graph retains dynamical memory rather than replacing it by a stationary Lambert value.

4.2 Attraction during passage

Lemma 4.1 (Finite-chart stable-direction estimate). In a sufficiently small fold chart the deviation $e=w-\psi(y,\lambda,\epsilon)$ decays exponentially for the time the complete trajectory stays in the chart, regardless of critical equilibrium stability.

Proof. Subtract the graph invariance equation at the actual $(y,\lambda,\epsilon)$. Parameter derivatives and the $B_fy^2$ term cancel:

$$\dot e=A_se+[N_w(y,\psi+e)-N_w(y,\psi)]-\psi_y[N_y(y,\psi+e)-N_y(y,\psi)].\tag{33}$$

The nonlinear derivatives in $w$ vanish at the origin and are $O(\delta)$ on a radius-$\delta$ chart; $\psi_y(0)=0$. Thus $\dot e=[A_s+O(\delta)]e+O(\lVert e\rVert^2)$. Choose $P_s>0$ on $E_s$ with $A_s^TP_s+P_sA_s=-I$. For small enough chart,

$$\frac d{dt}(e^TP_se)\le-\tfrac12\lVert e\rVert^2.\tag{34}$$

Norm equivalence gives $\lVert e(t)\rVert\le Ce^{-g(t-t_0)}\lVert e(t_0)\rVert$. Its effect on the critical equation is an integrable decaying perturbation. $\square$

After transients, variation of constants gives $w=O(\nu^{2/3})$ for $y=O(\nu^{1/3})$ and $\lambda=O(\nu^{2/3})$. If using moving stationary coordinates the forcing would include $-x_*'(a)\dot a$, and moving eigenvectors would add a connection term. The fixed chart avoids the singular branch derivative while keeping parameter motion in (31). Regular branch tracking below restores the moving-reference term explicitly.