Escape of a harmonically forced particle from an infinite-range potential well: a transient resonance

The paper considers a transient process of escape of a classical particle from a one-dimensional potential well. We address a particular model of the infinite-range potential well that allows independent adjustment of the well depth and of the frequency of small oscillations. The problem can be conveniently reformulated in terms of action-angle variables. Further averaging provides a nontrivial conservation law for the slow flow. Then, one can consider the problem in terms of averaged transient dynamics on primary 1:1 resonance manifold. This simplification allows efficient analytic exploration of the escape process. As a result, one obtains a theoretical prediction for minimal forcing amplitude required for the escape, as a function of the excitation frequency. This function exhibits a single sharp minimum for a certain intermediate frequency value, below the frequency of small free oscillations. This result conforms to earlier numeric and semi-analytic estimations for similar escape models, considered, in particular, in connection with problems of ship capsize and dynamic pull-in in microelectromechanical systems. The results presented in the paper allow conjecturing the generic dynamical mechanism, responsible for these regularities. In particular, the aforementioned sharp minimum in the frequency–amplitude domain is related to formation of heteroclinic connection between the saddle points on the resonance manifold. Numeric simulations are in complete qualitative and reasonable quantitative agreement with the theoretical predictions.