In this paper, we investigate the dynamics of particles within a bi-quartic potential well, characterized by the coupled potential function \( \displaystyle V(x, y) = \frac{1}{2}x^2 + \frac{1}{2}y^2 - \frac{1}{4}x^4 - \frac{1}{4}y^4 + Cx^2y^2 \) . Our focus is on the safe basins of escape and level-crossing under arbitrary initial conditions, i.e., the spatial region of initial conditions from where an initiated motion of the particle remains bounded. The coupling term allows energy exchange between the modes. If the total energy is sufficient, a particle starting from a given set of initial conditions within the potential well can reach the escape boundary over time, which would not occur without coupling. We find that escape trajectories often pass near one of the four saddles of the potential. Numerical simulations reveal that the safe basins of escape have fractal boundaries due to the energy-exchange mechanism. To address safety-critical applications where these chaotic regimes must be avoided, we introduce a factor of safety that defines a safety region. Crossing the safety region’s boundary shifts the problem from escape to level-crossing. Assuming harmonic-like solutions of the differential equations with slowly varying amplitudes and phases, 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 safe region boundary and calculate its size based on the coupling parameter C.
