Nonlinear analysis of electrostatic micro-electro-mechanical systems resonators subject to delayed proportional–derivative controller

Abstract

The present study deals with the nonlinear analysis of electrostatic micro-electro-mechanical systems resonators with two symmetric electrodes and subjected to delayed proportional–derivative controller. After a brief description of the model, the stability analysis of the linearized system is presented to depict the stability charts in the parameter space of proportional gain and time delay. The bifurcation diagram is used to confirm the existence of the delay-dependent and delay-independent regions and to analyze the effect of proportional–derivative gains and time delay on the dynamics of the system. Using Melnikov’s theorem, the criterion for the appearance of horseshoe chaos from homoclinic and heteroclinic bifurcations is presented. Melnikov’s predictions are confirmed by using the numerical simulations based on the basin of attraction of initial conditions. It is found that the increase in proportional gain contributes to increase the region of regular motion in both bifurcations. However, the increase in derivative gain contributes rather to reduce the region of regular motion for homoclinic bifurcation, although it increases rather this region in the case of heteroclinic bifurcation. Moreover, it is also observed, depending on proportional–derivative gains, the existence of a critical value of the delay where before it, the region of regular motion increases and after it, decreases rather.