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

Abstract

The paper presents an analytic 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 nonescaping 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.

Escape of a particle from two-dimensional potential well

“…This outer boundary is further eroded by the secondary resonances. Qualitatively, this result corresponds to the findings of the single-DOF model with harmonic external excitation [28].…”

“…These regions are assembled from small branches of weaker resonances which are not robust enough to seek appropriate analytical predictions. A similar phenomenon to the resonance sets obtained in this work was found in former research [28] that investigated the escape of particle from a potential well due to external forcing. In this case, for some ranges of the system parameter, the existence of secondary resonances governs the erosion of the safe basins, and hence decreases the prediction quality.…”

“…Using the characteristics of the dynamics in these regions, we try to predict the point of separation of the symmetric mode from the stability region, and the escape limits in its vicinity. Former work [25][26][27][28] employed the isolated resonance assumption to explore the escape of various systems mainly under external harmonic excitation. Some modifications are required to use this approach for the considered dynamical setting.…”

Safe basins of escape of a weakly-damped particle from a truncated quadratic potential well under harmonic excitation

“…Now, it is possible to insert the value of τ F (x 0 , u 0 ) back into Def. (23) and plot the boundary of the fast nonescaping set ∂N E F as an implicit function of (x 0 , u 0 ) (see Fig. 4 and 5).…”

“…The determination and understanding of the parameters' impact on the SB is of great engineering importance, contributing to the design of robust systems and applications. Karmi et al successfully derived a conservation law for a forced escape with the use of action angle variables and averaging for a quadratic-quartic well with truncation at different energy levels [23]; however, the fully analytic treatment of the SB is rarely possible due to the complexity of the problem. In the following, one of the exceptional cases, the escape of a weakly damped particle from a truncated quadratic potential well under harmonic excitation, is presented.…”