Non-generality of structural stability conjecture for high-dimensional systems

Let $f$ be a mapping (neural network) with sufficiently high dimension $d$, let $U$ be a bifurcation chain set as in the hyperbolicity violation conjecture, and let $V$ be the chain link set. Let $C_{\max}$ be the largest connected component of $\overline V$. The non-generality of structural stability conjecture asserts that the perturbation size $\delta_s$ of $s\in C_{\max}$ for which $f|_{C_k}$ remains structurally stable tends to zero as $d\to\infty$. The paper contrasts this claim with classical examples and discusses whether it extends to high-dimensional parameter surfaces; no resolution is supplied.