Summary
In this paper, we investigate the dynamics of particles within a bi-quartic potential well, characterized by the coupled potential function \( V(x, y) = \frac{1}{2}x^2 + \frac{1}{2}y^2 - \frac{1}{4}x^4 - \frac{1}{4}y^4 + Cx^2y^2 \). Our focus is on the safe basins of escape (SBoE) and level-crossing (SBoLC) within the initial displacement conditions (IDCs) plane, where the particle's motion is bounded. The study is limited to uniform initial velocity conditions, although extending it to arbitrary initial conditions (ICs) is straightforward. The coupling term allows energy exchange between the modes, enabling IDCs with enough energy to potentially escape the well over time. We find that escape trajectories often pass near one of the four saddles of the potential. Numerical simulations reveal that the SBoE has fractal boundaries because of the energy exchange mechanism. To address safety-critical applications where these chaotic regimes must be avoided, we introduce a factor of safety (FoS) that leads to the definition of a safety region (SR). Crossing the SR's boundary shifts the problem from escape to level-crossing (LC). Assuming harmonic-like solutions of the differential equations, which have slowly varying amplitude and phase values, we transform our system into an appropriate form for averaging. By eliminating time as a variable and realizing that only the phase difference is significant, we derive two first integrals of the particle motion in analytic form, which allows us to analytically determine the SR boundary and calculate its size based on the coupling parameter $C$.
