An Exact Bistability Threshold on Three Nodes

5. Three-node path: exact count and threshold

5.1 Modes and scalar stationary law

For $b_1=(1,-1,0)^T$, $b_2=(0,1,-1)^T$ the orthonormal eigenbasis is

$$v_0=\frac{(1,1,1)^T}{\sqrt3},\quad v_1=\frac{(1,0,-1)^T}{\sqrt2},\quad v_3=\frac{(1,-2,1)^T}{\sqrt6},\qquad Lv_\lambda=\lambda v_\lambda.\tag{26}$$

For $f=av_\lambda$, $a>0$ and $\lambda\in\{1,3\}$, equal-memory stationary states have $z=uv_\lambda$, $m_1=m_2=-s$, and $|b_e^Tz|^2=\lambda|u|^2/2$. Thus

$$R^2=|u|^2=\frac{4\kappa}{\lambda}se^s,\qquad u=\frac{a/c}{\mu+\lambda e^{-s}}.\tag{27}$$ $$\frac{a^2}{|c|^2}=H_\lambda(s):=\frac{4\kappa}{\lambda}se^s(\mu+\lambda e^{-s})^2.\tag{28}$$

The input fixes phase through $a/c$. Signed reflection yields the low-mode sector, ordinary reflection the highest-mode sector, with zero collective stationary amplitude. Differentiation gives

$$H_\lambda'(s)=\frac{4\kappa}{\lambda}(\mu+\lambda e^{-s})e^s(1+s)[\mu-\lambda Q(s)].\tag{29}$$

If $\mu\ge\lambda\chi$, the law increases strictly from zero to infinity; equality permits one isolated flat point. If $0<\mu<\lambda\chi$, the two roots of $Q(s)=\mu/\lambda$ obey

$$1<s_-<\sqrt3<s_+.\tag{30}$$

The first gives a strict local maximum, the second a strict local minimum. Set

$$a_{\rm down}=|c|\sqrt{H_\lambda(s_+)},\qquad a_{\rm up}=|c|\sqrt{H_\lambda(s_-)}.\tag{31}$$

The three monotonicity intervals give three distinct positive roots in the open interval, two distinct roots including a double root at either endpoint, and one outside the closed interval. At an equal-edge equilibrium, rotate the common phase to make amplitude real. The radial and tangential Schur operators are $\mu I-Q(s)L$ and $\mu I+e^{-s}L$:

$$\mu,\quad\mu-Q,\quad\mu-3Q;\qquad \mu,\quad\mu+w,\quad\mu+3w,\qquad w=e^{-s}.\tag{32}$$

The two collective quadratures decay, since the mean $\bar z=(z_1+z_2+z_3)/3$ obeys

$$\dot{\bar z}=-c\mu\bar z+\bar f.\tag{33}$$

5.2 Sector stability can miss a full instability

At a low-mode fold $Q=\mu$, the transverse highest-mode radial curvature is $-2\mu<0$. A retained exact example takes $\mu=0.01$, $\kappa=1$, $s=5$, $a=|c|\sqrt{H_1(5)}$. Here $Q(5)=\tfrac23e^{-5}$ and rational exponential bounds give $1/300<Q(5)<1/100$. The sector radial curvature is approximately $0.00550804$, while the transverse curvature is $-0.00347589$. The full Jacobian has one unstable eigenvalue despite a positive clamped carrier. This rejects the candidate, without ruling out all low-mode bistability: two sufficiently far outer low-mode states can have positive full curvature.

5.3 Complete count for highest-mode preparation

Theorem 5.1 (Complete path bistability). For $\chi<\mu<3\chi$ and $f=av_3$, $a>0$, (27)–(28) with $\lambda=3$ give all equilibria. The open fold interval contains two locally exponentially stable minima and one index-one saddle. Outside the closed interval there is one globally asymptotically stable equilibrium. Each colliding pair at an endpoint has one zero and seven decaying linear modes. Every constant-input trajectory converges to a stationary point in these finite regimes.

Proof. Write $z=tv_0+uv_3+hv_1$ with complex $t,u,h$. Edges and forcing do not see $t$; the collective quadratic forces $t=0$ at stationarity. Reflection of nodes 1 and 3 fixes $v_3$, reverses $v_1$, and interchanges edges up to irrelevant signs. At fixed $u$ the potential is even in $h$, and the $v_1$ eigenvalue of $L$ gives

$$\nabla_h^2V_{\rm eff}\ge(\mu-\chi)I_2>0.\tag{34}$$

Thus $h=0$ is its unique critical point. Both edge magnitudes then coincide, so (11) gives $m_1=m_2=-s$. The stationary amplitude equation fixes $u=(a/c)/(\mu+3e^{-s})$, yielding exactly (28). The scalar count is therefore complete. Outer branches have $Q<\mu/3$; (32), (21), and Lemma 4.1 give local exponential stability. The middle branch has $\mu/3<Q\le\chi<\mu$, so exactly one Schur direction is negative and the full Jacobian has one right-half-plane eigenvalue. At a fold $Q=\mu/3$ and all other Schur directions are positive, giving a simple zero and seven decaying modes. Proposition 2.1 gives convergence in all finite cases. In the one-root case coercivity gives a strict global minimum of $V$, hence Lyapunov stability as in Theorem 3.1. This last conclusion is not asserted for the nonhyperbolic colliding point. $\square$

Corollary 5.2 (Sharp path threshold and optimal universal coefficient). Every constant complex input on this path has exactly one globally asymptotically stable equilibrium if and only if $\mu\ge3\chi$. Consequently $\chi$ is the smallest universal sufficient coefficient $C$ in $\mu\ge C\lVert L\rVert$.

Proof. $\lVert L\rVert=3$, so Theorem 3.1 gives sufficiency. For every $0<\mu<3\chi$, choose highest-mode load strictly between the folds. Both outer scalar roots obey $Q<\mu/3$ and are fully stable by (32) and Lemma 4.1. This disproves all-input uniqueness even for $\mu\le\chi$, where the entire stationary set has not been counted. If $C<\chi$, choose positive $\mu<3\chi$ with $\mu\ge3C$ to disprove that universal coefficient. $\square$

These are distinct statements: a universal sufficient condition, its optimal coefficient over networks, an exact all-input threshold on one path, and uniqueness at one specified drive. At fixed physical $a$ the three-state condition in the counted range is exactly

$$H_3(s_+)<\frac{a^2}{\gamma^2+\alpha^2}<H_3(s_-).\tag{35}$$

Equivalently, $a^2/H_3(s_-)-\gamma^2<\alpha^2<a^2/H_3(s_+)-\gamma^2$, restricted to $\alpha>0$. At a positive endpoint the load derivative $-2\alpha a^2/(\gamma^2+\alpha^2)^2$ is nonzero, so mobility can pass through a fold at fixed physical drive. This changes the stationary Hessian and does not contradict the fixed-$H$ result. The stationary equations do not contain $\eta$.