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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 consider a system of coupled partial differential equations modeling the interaction of mussels and algae in advective environments. A key parameter in the equations is the ratio of the diffusion rate of the mussel species and the advection rate of the algal concentration. When advection dominates diffusion, one observes large-amplitude solutions representing bands of mussels propagating slowly in the upstream direction. Here, we prove the existence of a family of such periodic wavetrain solutions. Our proof relies on geometric singular perturbation theory to construct these solutions as periodic orbits of the associated traveling wave equations in the large-advection{small-diffusion limit. The construction encounters a number of mathematical obstacles which necessitate a compactification of phase space, geometric desingularization to remedy a loss of normal hyperbolicity, and the application of a generalized exchange lemma at a loss-of-stability turning point. In particular, our analysis uncovers logarithmic (switchback) corrections to the leading-order solution. [ABSTRACT FROM AUTHOR]

Mixed-mode dynamics is a complex type of dynamical behavior that is characterized by a combination of small-amplitude oscillations and large-amplitude excursions. Mixed-mode oscillations (MMOs) have been observed both experimentally and numerically in various prototypical systems in the natural sciences. In the present article, we propose a mathematical model problem which, though analytically simple, exhibits a wide variety of MMO patterns upon variation of a control parameter. One characteristic feature of our model is the presence of three distinct time-scales, provided a singular perturbation parameter is sufficiently small. Using geometric singular perturbation theory and geometric desingularization, we show that the emergence of MMOs in this context is caused by an underlying canard phenomenon. We derive asymptotic formulae for the return map induced by the corresponding flow, which allows us to obtain precise results on the bifurcation (Farey) sequences of the resulting MMO periodic orbits. We prove that the structure of these sequences is determined by the presence of secondary canards. Finally, we perform numerical simulations that show good quantitative agreement with the asymptotics in the relevant parameter regime. [ABSTRACT FROM AUTHOR]