2018/07/05 by Iuorio, Annalisa, Popovic, Nikola, Szmolyan, Peter
#34B16 #34C23 #34E05 #34E15 #34L30 #35K67 #74G10 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1807.02069
Micro-Electro Mechanical Systems (MEMS) are defined as very small structures that combine electrical and mechanical components on a common substrate. Here, the electrostatic-elastic case is considered, where an elastic membrane is allowed to deflect above a ground plate under the action of an electric potential, whose strength is proportional to a parameter λ. Such devices are commonly described by a parabolic partial differential equation that contains a singular nonlinear source term. The singularity in that term corresponds to the so-called "touchdown" phenomenon, where the membrane establishes contact with the ground plate. Touchdown is known to imply the non-existence of steady state solutions and blow-up of solutions in finite time. We study a recently proposed extension of that canonical model, where such singularities are avoided due to the introduction of a regularizing term involving a small "regularization" parameter ε. Methods from dynamical systems and geometric singular perturbation theory, in particular the desingularization technique known as "blow-up", allow for a precise description of steady-state solutions of the regularized model, as well as for a detailed resolution of the resulting bifurcation diagram. The interplay between the two main model parameters ε and λ is emphasized; in particular, the focus is on the singular limit as both parameters tend to zero.