On a bi-stability regime and the existence of odd subharmonics in a Comb-drive MEMS model with cubic stiffness

In this paper we study two different problems. First we present a novel result about the existence of a family of odd subharmonics with prescribed nodal properties for a general nonlinear oscillator with bounded domain and symmetries. Then we apply the general existence result to the Comb-drive finger MEMS model with a cubic nonlinear stiffness term, and prove analytically that the odd positive subharmonic of order two is linearly stable whenever the AC load of the input voltage is small enough. Moreover, we demonstrate a bi-stability regime for this model because the trivial solution $x\equiv 0$ is also linearly stable. The general existence result was obtained through a generalization of the Ortega’s variational principle (Ortega, 2016), and the stability assertions were obtained by following the perturbation approach in Cen et al. (2020) for the linear stability of a nontrivial periodic solution that emanates from the autonomous problem, and the well-known Zukovskii criterion.