In this paper, we study a canonical mass-spring electrostatically actuated Micro electro-mechanical systems (MEMS) model. We investigate the Ambrosetti-Prodi bifurcation phenomena of periodic solutions and analyze the dynamical behavior as the parameter varies continuously. The main results partially address the open problem posed by Torres and provide theoretical support for the stability analysis and design of MEMS devices. Further, through numerical bifurcation analysis, we provide bifurcation diagrams, time series and phase diagrams to reveal the key results.

In this paper, we first gave the Green's function of a sixth-order linear differential equation with variable coefficients, and then proved the existence of positive periodic solutions for a sixth-order differential equation with a singularity of repulsive type via Schauder's fixed point theorem.

This paper is devoted to studying the existence of at least one periodic solution for a generalized Basener-Ross model with time-dependent coefficients. The discussion is based on the Man\’asevich-Mawhin continuation theorem and fixed point theorem of cone mapping together with some properties of Green’s function.

By application of Leray-Schauder alternative principle and Krasnoselskii's fixed point theorem on compression and expansion of cones, existence results of positive periodic solutions are presented for a second-order neutral differential equation with singularity of repulsive type.