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

Various models and systems involving the escape of periodically forced particle from the potential well demonstrate a common pattern. 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 detailed exploration of the transient dynamics of the escape from a number of benchmark potential wells. In the truncated parabolic well, in the absence of the damping, the minimal forcing amplitude required for the escape obviously tends to zero, as the excitation frequency approaches the natural frequency of oscillations in the well. Perturbation of the parabolic truncated well by weak symmetric softening nonlinearity leads to a shift of the minimum forcing to a nonzero value at the frequency below the natural. We explicitly compute this shift in the principal approximation by considering the slow-flow dynamics in conditions of the principal 1:1 resonance. The addition of nonlinearity also leads to appearance of two qualitatively different scenarios of the escape transition. In the first scenario, the slow flow just approaches the maximum value required for the escape. In the second scenario, the slow flow approaches dynamical saddle point that appears due to the external forcing. The presence of two competing and qualitatively different escape scenarios makes it impossible to formulate empirical escape criteria based on the steady-state solutions. Essentially, nonlinear phi(4) model requires slightly more complex mathematical tools (transformation to action-angle variables), but demonstrates very similar qualitative features of the transient dynamics, including two competing scenarios of the escape transition.