SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2749-2782, January 2020.
We study a nonlinear wave equation appearing as a model for a membrane (without viscous effects) under the presence of an electrostatic potential with strength $\lambda$. The membrane has a unique stable branch of steady states $u_{\lambda}$ for $\lambda\in\lbrack0,\lambda_{\ast}]$. We prove that the branch $u_{\lambda}$ has an infinite number of branches of periodic solutions (free vibrations) bifurcating when the parameter $\lambda$ is varied. Furthermore, using a functional setting, we compute numerically the branch $u_{\lambda}$ and its branches of periodic solutions. This approach is useful to validate rigorously the steady states $u_{\lambda}$ at the critical value $\lambda_{\ast}$.

中文翻译：

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MEMS波动方程建模中的自由振动

SIAM应用动力系统杂志，第19卷，第4期，第2749-2782页，2020年1月。
我们研究了非线性波动方程，该波动方程作为膜的模型（无粘性效应）在存在静电势强度为\\的情况下出现lambda $。膜具有唯一的稳态稳定分支，分别为$ \ lambda \ in \ lbrack0，\ lambda _ {\ ast}] $。我们证明，当参数$ \ lambda $变化时，分支$ u _ {\ lambda} $具有无限数量的周期解（自由振动）分支。此外，使用函数设置，我们以数字方式计算分支$ u _ {\ lambda} $及其周期解的分支。此方法对于严格验证处于临界值$ \ lambda _ {\ ast} $的稳态$ u _ {\ lambda} $很有用。