Bound on Amplitude of a MEMS Resonator by Approximating the Derivative of the Lyapunov Function in Finite Time

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 Omega = 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.