Characterising chaos in a multi-magnetic spherical pendulum: A computational approach using Lyapunov analysis

By implementing both numerical and computational methods, we illustrate the degree of chaos present in a magnetically influenced spherical pendulum by using dipole-like interactions between the pendulum mass and surrounding magnets. When the magnets produce a repulsive force which is inversely proportional to the fourth power of the distance, the system is observed to be chaotic such that it demonstrates sensitivity to initial conditions. By increasing the initial azimuthal velocity, 𝜙̇ , in initial condition vector [𝜃0, 𝜙0, 𝜃̇0,𝜙̇0], one can observe the increased level of chaos within the system. This is quantified using the Maximum Lyapunov Exponent, MLE, which shows a positive value of 𝜆𝑚𝑎𝑥 > 0 for all measured initial conditions. This value ranges from 𝜆𝑚𝑎𝑥~0.15 for lower energy conditions, to 𝜆𝑚𝑎𝑥~0.7 for higher energy conditions.