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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 \lambda. 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 varepsilon. 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 varepsilon and \lambda is emphasized; in particular, the focus is on the singular limit as both parameters tend to zero. [ABSTRACT FROM AUTHOR]

We study the existence of profiles for Chapman-Jouguet detonation and deflagration waves in the Navier-Stokes equations for a reacting gas. In the limit of small viscosity, heat conductivity, and diffusion, the profiles correspond to heteroclinic orbits of a system of singularly perturbed ordinary differential equations. The burned end state of the waves, however, is a nonhyperbolic equilibrium of the associated, purely gas dynamic layer problem, and hence standard methods from geometric singular perturbation theory fail. We show how to resolve this degeneracy by combining a center manifold reduction with the blow-up method. The main result is the existence of viscous profiles for various types of Chapman-Jouguet processes. In addition, we obtain results on the spatial decay rates of these waves which are expected to be relevant for the stability analysis of the waves. [ABSTRACT FROM AUTHOR]

A detailed geometric analysis of the Goldbeter-Lefever model of glycolytic oscillations is given. In suitably scaled variables the governing equations are a planar system of ordinary differential equations depending singularly on two small parameters ε and δ. In [L. Segel and A. Goldbeter, J. Math. Biol., 32 (1994), pp. 147-160] it was argued that a limit cycle of relaxation type exists for ε « δ « 1. The existence of this limit cycle is proved by analyzing the problem in the spirit of geometric singular perturbation theory. The degeneracies of the limiting problem corresponding to (ε, δ) = (0, 0) are resolved by a novel variant of the blow-up method. It is shown that repeated blowups lead to a clear geometric picture of this fairly complicated two-parameter multiscale problem. [ABSTRACT FROM AUTHOR]

We construct self-similar solutions of the Dafermos regularization of a system of conservation laws near structurally stable Riemann solutions composed of Lax shocks and rarefactions, with all waves possibly large. The construction requires blowing up a manifold of gain-of-stability turning points in a geometric singular perturbation problem as well as a new exchange lemma to deal with the remaining hyperbolic directions. [ABSTRACT FROM AUTHOR]