## On the stability of periodic solutions with defined sign in MEMS via lower and upper solutions.(English)Zbl 1464.34062

Summary: This article considers the existence and linear stability of positive periodic solutions for the Nathanson’s model and the Comb-drive finger model of Micro-Electro-Mechanical-Systems (MEMS). Our results contribute to a better understanding of these models and therefore, they add up to the set of tools to decide the range of usefulness in sensing and acting of these two models. Both models are considered without damping and having an AC-DC input voltage $$\mathcal{V}(t) > 0$$ of period $$T^\ast > 0$$. For the Nathanson’s model, we prove the existence of two $$T^\ast$$-periodic solutions $$0 < \varphi(t) < \psi(t)$$ under the assumption that the voltage is less than the pull-in voltage and greater than a certain positive constant depending on the temporal frequency of $$\mathcal{V}$$ and physical parameters. The solution $$\varphi$$ is elliptic and, if the parameter $$\lambda := \frac{\min \mathcal{V}}{\max \mathcal{V}}$$ is greater than a certain critical value, the solution $$\psi$$ is hyperbolic. We point out that if $$\mathcal{V}$$ is greater than the pull-in voltage, there are not $$T^\ast$$-periodic solutions. Under the same conditions that guarantee the existence, we prove the uniqueness of these two solutions and we classify their linear stability. We can view this existence/uniqueness/linear stability classification theorem as an extension of the known results for the autonomous case when the voltage $$\mathcal{V}$$ is constant. The Comb-drive model is studied under a cubic stiffness with coefficient $$\alpha > 0$$, and its dynamics depends on whether $$\alpha \leq 2$$ or $$\alpha > 2$$. We will prove that for an adequate range of $$\mathcal{V}$$ the equilibrium (the zero solution) is elliptic. This suggests the existence of periodic solutions with defined sign. In this document we prove the existence of periodic positive (negative) solutions for the Comb-drive model for which the number and the linear stability depend on $$\alpha$$ and $$\lambda$$ respectively. When $$0 < \alpha \leq 2$$, we prove the existence of an unstable periodic solution for $$\lambda$$ greater than a certain critical value and $$\mathcal{V}$$ below the pull in voltage. For the case $$\alpha > 2$$, similar to the Nathanson case, we prove the existence of two positive periodic solutions one of which is linearized stable and the other is unstable. The methodology uses the Lower and Upper Solution Method. Some numerical examples are provided to illustrate the results.

### MSC:

 34C25 Periodic solutions to ordinary differential equations 34D20 Stability of solutions to ordinary differential equations 70K40 Forced motions for nonlinear problems in mechanics
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### References:

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