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Journal of Applied Nonlinear Dynamics 11(1) (2022) 247--269 | DOI:10.5890/JAND.2022.03.015

Andr 'es Rivera , Jeremy J. Thibodeaux, Johan S. S 'anchez

Department of Natural Sciences and Mathematics, Pontificia Universidad Javeriana Cali, 760031, Cali-Colombia

Department of Mathematical Sciences, University of the Virgin Islands

Department of Mathematics and Computer Science, Loyola University New Orleans, New Orleans, LA, USA

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Abstract

The purpose of this article is to analytically prove the exis\-tence and linear stability of three $\hat{T}$-periodic solutions (two strictly positive and one strictly negative) for a comb-drive actuator where the moveable finger is initially shifted a small distance $u>0$ from the center of the two fixed ones. Here we assume a damping force that is proportional to the velocity and an $AC$-$DC$ driving voltage $\hat{V}(\tau)>0$ with period $\hat{T}>0$. Under appropriate conditions over $\hat{V}_{\min}:=\min \hat{V}(\tau)$ and $\hat{V}_{\max}:=\max V(\tau)$, one of these solutions will be elliptic and the other two are hyperbolic. The basic tools for proving our results are the Lower and Upper Solution Method, Degree Theory and the Lyapunov-Zukovskii stability criterion for Hill's equation. Some numerical examples are provided to illustrate the results.

Acknowledgments

The authors A. Rivera and J. S. S\'anchez have been financially su\-pported by Convocatoria Interna project (2019) No.\,1551. The author J. Thibo\-deaux has been financially supported by a grant from the Fulbright U.S. Scholar program.

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