Dynamical integrity of the safe basins in a problem of forced escape

Abstract

This paper explores the use of the Approximation of Isolated Resonance (AIR) method for determining the safe basins (SBs) in the problem of escape from a potential well. The study introduces a novel approach to capture the location and the shape of the SBs and establish their erosion profiles. The research highlights the concept of "true" safe basins, which remain invariant with phase shifts, a critical factor often faced in real-world applications. A cubic polynomial potential serves as the benchmark to illustrate the proposed method.

Figures (7)

• Figure 1: Passing through the saddle-connection on the $(\vartheta,\xi)$-cylinder. Blue and red curves are the level sets defined by the maximum and the saddle mechanisms, respectively. Dashed lines represent the boundaries of the "true" SBs. The energy threshold is $\xi_{\max}=0.1657$ and the external frequency is $\Omega=0.89$
• Figure 2: Panel (a): Approximation of the erosion profile for a truncation level $\xi=0.1567$. Panel (b): Critical value $\widehat{F}$ as a function of the excitation frequency $\Omega$. Dashed and solid curves correspond to $\xi_{\max}=0.1657$ and $\xi_{\max}=0.155$, respectively
• Figure 3: Actual escaping ICs (yellow) plotted on the $(\vartheta, \xi)$-cylinder
• Figure 4: Panel (a): Black dots represent numerically obtained thresholds that account for the stochastic layer. Gray dashed line is the maximum depth of the potential well, i.e., $\xi=1/6$. Panel (b): Erosion profiles, i.e., SB areas depending on the forcing amplitude $F$. Black curve represents the area of SBMT. Red curve is the area of SBST (dashed part denotes area of SBST which is completely included inside SBMT). Black dots correspond to the area of SB computed numerically. Grey curve represents the sum of the areas of the SB of both types.
• Figure 5: Evolution of the safe basin for the phase $\psi=0$. Gray color represents the initial conditions for which the numerical simulations show no escape. Each numerical simulation runs for $100$ periods of the external excitation with the frequency $\Omega=0.89$