Quantitative Analysis of Basin-Boundary Complexity in Multi-Attractor Magnetic Pendulum Systems

This study investigates the influence of magnet configuration on basin-boundary complexity in a simplified magnetic-pendulum-inspired system. The model considers a freely moving particle in a two-dimensional plane under the influence of multiple fixed magnetic attractors. Numerical simulations were performed by varying the number of attractors, N, from 2 to 5 and the radial distance, R, of the attractors from the centre of the system. For each configuration, basin-of-attraction maps were generated using a uniform grid of initial conditions, and a boundary-ratio metric was used to quantify the degree of basin complexity. The results show that increasing the number of attractors generally increases basin-boundary complexity and sensitivity to initial conditions. The N=2 system produced simple basin structures with a single dominant boundary, whereas systems with N=3, N=4, and N=5 exhibited increasingly interwoven and fragmented boundaries. However, maximum boundary complexity did not occur at the smallest radial spacing. Instead, the highest boundary ratios were observed at intermediate values of R, indicating that basin complexity is controlled by a balance between attractor separation and interaction strength. Nonlinear regression further indicated that the relationship between R and boundary complexity follows a systematic nonlinear trend. These findings suggest that attractor geometry plays a critical role in controlling predictability and basin-boundary formation in multi-attractor magnetic pendulum systems.