Nonvanishing parameter derivative at the dynamic threshold

Prove that the signed distance from the stable manifold of the hyperbolic saddle to a fixed transversal has a nonvanishing derivative with respect to the electrostatic parameter α at α = αdyn, thereby validating the coefficient −1/λ+ in the logarithmic pull-in-time asymptotic law of Eq. (B19).

Background

For damping below the critical value, the from-rest trajectory reaches a hyperbolic saddle at the dynamic pull-in threshold, producing logarithmic divergence of the pull-in time. The coefficient in Eq. (B19) assumes that the distance between the trajectory and the saddle’s stable manifold varies linearly with α near αdyn. The paper establishes smooth dependence but explicitly does not establish that the derivative is nonzero; proving this transversality would rule out higher-order vanishing and confirm the stated coefficient rather than a modified coefficient −k/λ+.