= Solution
Averaging the fast fluctuations in $yx=xx^Tw$ gives
$$
\tau\dot w=Cw-|w|^2w,\qquad C=\langle xx^T\rangle.
$$
Thus a nonzero steady <vector> satisfies
$$
\boxed{Cw=\lambda w,\qquad\lambda=|w|^2}.
$$
Its direction is an <eigenvector> of the input second-moment <matrix>, and its magnitude is the square root of that <eigenvalue>. The zero <vector> is also a steady state and satisfies the displayed relation with $\lambda=0$. This is an averaged steady state; the microscopic fluctuating update need not vanish at each instant.
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