Particle Escape from a Potential Well Under High-Frequency External Forcing

This paper studies the escape dynamics of a periodically forced particle from a potential well. The excitation frequency is assumed to be substantially higher than the natural frequency of the potential well at the stable equilibrium. At this stage, damping is neglected, and the quadratic-quartic Duffing potential oscillator is considered. The paper analyzes two main problems: (i) The escape threshold – for fixed initial conditions (IC), the combinations of the excitation parameters that correspond to the escape transition are determined. (ii) The safe basin – for a given set of excitation parameters, the set of the non-escaping IC is determined. The analytical solution is obtained as a sum of fast and slow components and is compared to the numerical results. The first primary problem is determining the critical forcing amplitude as a function of the excitation frequency. Approximations of the amplitude equation are performed for different phases and zero IC. The results show good correspondence between the numerical and analytical approximations. For the safe basin problem, we analytically demonstrate that the boundary of the non-escaping IC region is a union of two parabolic fragments. ... mehrIn this case, significant discrepancies between the numerical and analytical solutions are identified for extreme forcing values. By analyzing the causes of these differences, we aim to gain a deeper understanding of the dynamics of the Duffing oscillator.