Basic mechanisms of escape of a harmonically forced classical particle from a potential well

TL;DR

This paper investigates how a classical particle escapes from a potential well under periodic forcing, revealing the influence of nonlinearity and resonance on the minimal forcing amplitude needed for escape.

Contribution

It explains the regularity of escape thresholds using simple models and analyzes the effects of nonlinearity and resonance on escape dynamics.

Findings

Minimal forcing amplitude tends to zero at natural frequency in linear case.

Weak nonlinearity introduces a nonzero forcing minimum below the natural frequency.

Qualitative features of escape dynamics are similar across different nonlinear models.

Abstract

In various models and systems involving the escape of periodically forced particle from the potential well, a common pattern is observed. Namely, the minimal forcing amplitude required for the escape exhibits sharp minimum for the excitation frequency below the natural frequency of small oscillations in the well. The paper explains this regularity by exploring the transient escape dynamics in simple benchmark potential wells. In the truncated parabolic well, in absence of the damping the minimal forcing amplitude obviously tends to zero for the natural excitation frequency. Addition of weak symmetric softening nonlinearity to the truncated parabolic well leads to the nonzero forcing minimum below the natural frequency. We explicitly compute this shift in the principal approximation by considering the slow-flow dynamics in conditions of the principal 1:1 resonance. Essentially nonlinear model, analyzed with the help of transformation to action-angle variables, demonstrates very similar qualitative features of the transient escape dynamics.