Approximation of potential function in the problem of forced escape

TL;DR

This paper develops a method to approximate realistic potentials with low-order polynomials for better analysis of escape dynamics in forced systems, demonstrated on a MEMS pull-in problem.

Contribution

It introduces a polynomial approximation approach for potential functions to facilitate action-angle analysis in escape problems, applicable to complex realistic potentials.

Findings

Global L^2-optimal polynomial approximation yields the most accurate escape threshold predictions.

The method effectively models electrostatic potentials in MEMS systems for dynamic pull-in analysis.

Approximation improves the analytical description of escape dynamics in forced oscillatory systems.

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.