We propose a methodology to obtain the amplitude of a nonlinear differential equation that may not satisfy Lyapunov’s global stability criterion. This theory is applied to the MEMS resonator which has a high-quality factor. The derivative of the Lyapunov function approximated for a finite time and an optimization problem was formulated. The local optima were obtained using the Karush–Kuhn–Tucker conditions, for which the amplitude was analytically formulated. The obtained amplitude, when compared with that by the numerical method, showed the validity of the analytical approximation for a useful range of the nonlinearity, but accurate only at an excitation frequency Ω=0.913. This methodology will be useful to approximate the damping in a system if one obtains the amplitude from the experimental data near this excitation frequency.

中文翻译：

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通过在有限时间内逼近李雅普诺夫函数的导数来限制 MEMS 谐振器的幅度

我们提出了一种方法来获得可能不满足 Lyapunov 全局稳定性准则的非线性微分方程的幅度。该理论适用于具有高品质因数的MEMS谐振器。Lyapunov函数的导数在有限时间内逼近，并提出了一个优化问题。使用 Karush-Kuhn-Tucker 条件获得局部最优值，对其振幅进行了解析公式化。与数值方法相比，获得的幅度表明解析近似在非线性有用范围内的有效性，但仅在激励频率下才准确Ω=0.913. 如果从该激励频率附近的实验数据中获得幅度，则该方法将有助于近似系统中的阻尼。