Escape of a forced-damped particle from weakly nonlinear truncated potential well

TL;DR

This paper investigates how viscous damping affects the escape of a harmonically forced particle from a weakly nonlinear potential well, revealing different escape mechanisms compared to the linear case.

Contribution

It provides an explicit analytic evaluation of the damping's effect on escape thresholds using multiple-scales analysis for the weakly nonlinear case.

Findings

Damping influences escape thresholds explicitly.

Escape mechanisms differ significantly between linear and weakly nonlinear cases.

Slow-flow equations are non-integrable with damping, but analyzable for small damping.

Abstract

Escape from a potential well is an extreme example of transient behavior. We consider the escape of the harmonically forced particle under viscous damping from the benchmark truncated weakly nonlinear potential well. Main attention is paid to most interesting case of primary 1:1 resonance. The treatment is based on multiple-scales analysis and exploration of the slow-flow dynamics. Contrary to Hamiltonian case described in earlier works, in the case with damping the slow-flow equations are not integrable. However, if the damping is small enough, it is possible to analyze the perturbed slow-flow equations. The effect of the damping on the escape threshold is evaluated in the explicit analytic form. Somewhat unexpectedly, the escape mechanisms in terms of the slow flow are substantially different for the linear and weakly nonlinear cases.