Rigorous proof of the center-manifold pull-in-time divergence law

Prove rigorously that, for damping ratios above the critical value at which the from-rest dynamic threshold coincides with the static fold, the pull-in time satisfies the inverse-square-root asymptotic law in Eq. (B18) as the electrostatic parameter approaches the fold threshold from above, including the stated coefficient.

Background

The paper distinguishes two asymptotic regimes for the pull-in time. When the damping ratio exceeds the critical value, the from-rest trajectory approaches the saddle-node fold along its center manifold, and the authors derive the inverse-square-root law τPI = K(α − αc)−1/2. However, the derivation is based on a dominant-balance and matched-scaling argument rather than a complete proof. Establishing this result rigorously would complete the mathematical justification of the high-damping threshold asymptotics and substantiate the transition from logarithmic divergence below the critical damping to inverse-square-root divergence above it.