Analytic exploration of safe basins in a benchmark problem of forced escape

Abstract

The paper presents an analytical approach for predicting the safe basins (SB) in a plane of initial conditions (IC) for escape of classical particle from the potential well under harmonic forcing. The solution is based on the approximation of isolated resonance, which reduces the dynamics to conservative flow on a two-dimensional resonance manifold (RM). Such a reduction allows easy distinction between escaping and non-escaping ICs. As a benchmark potential, we choose a common parabolic-quartic well with truncation at varying energy levels. The method allows accurate predictions of the SB boundaries for relatively low forcing amplitudes. The derived SBs demonstrate an unexpected set of properties, including decomposition into two disjoint zones in the IC plane for a certain range of parameters. The latter peculiarity stems from two qualitatively different escape mechanisms on the RM. For higher forcing values, the accuracy of the analytic predictions decreases to some extent due to the inaccuracies of the basic isolated resonance approximation, but mainly due to the erosion of the SB boundaries caused by the secondary resonances. Nevertheless, even in this case the analytic approximation can serve as a viable initial guess for subsequent numeric estimation of the SB boundaries.