= Solution
Since the derivative of $\tanh(\beta x)$ at zero is $\beta$, the <Jacobian matrix> of this directed rate network is
$$
J=\frac1\tau\begin{pmatrix}-1&0&\beta W_{13}\\\beta W_{21}&-1&0\\0&\beta W_{32}&-1\end{pmatrix}.
$$
With $a=1+\tau\lambda$, its characteristic determinant is
$$
\tau^3\det(\lambda I-J)=\det\begin{pmatrix}a&0&-\beta W_{13}\\-\beta W_{21}&a&0\\0&-\beta W_{32}&a\end{pmatrix}=a^3-\beta^3W_{13}W_{32}W_{21}.
$$
Thus the <eigenvalues> satisfy
$$
\boxed{(1+\tau\lambda)^3=\Gamma,\qquad\Gamma=\beta^3W_{13}W_{32}W_{21}}.
$$
This establishes <three-neuron ring stability> from the loop product. The ring is directed, so the symmetric-weight <Hopfield network> energy argument does not apply. Assume the physical relaxation time $\tau>0$.
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