Approximation of potential function in the problem of forced escape

Abstract

The paper addresses an escape of a classical particle from a potential well under harmonic forcing. Most dangerous/efficient escape dynamics reveals itself in conditions of 1:1 resonance and can be described in the framework of isolated resonant (IR) approximation. The latter requires reformulation of the problem in terms of action-angle (AA) variables, available only for a handful of the model potentials. The paper suggests approximation of realistic generic potentials by low-order polynomial functions, admissible for the AA transformation, with possible truncation. To illustrate the idea, we first formulate the AA transformation and solve the escape problem in the IR approximation for a generic quartic potential. Then, the model problem for dynamic pull-in in microelectromechanical system (MEMS) is analyzed. The model electrostatic potential is approximated by the quartic polynomials (globally and locally), and quality of predicting the escape thresholds is assessed numerically. Most accurate predictions are delivered by global $L^2$-optimal heuristic approximation.