Comprehensive analysis of the mechanism of sensitivity to initial conditions and fractal basins of attraction in a novel variable-distance magnetic pendulum

Abstract

A novel variable-distance magnetic pendulum model is proposed to explore its sensitivity to the initial conditions. In this study, equilibrium points and stability theorems were employed as preliminary quantitative methods to determine the existence of multiple coexisting attractors in the model. Furthermore, the trajectories of the pendulum and time histories were applied as qualitative methods to demonstrate the sensitivity to the initial conditions. Finally, the evolution of the basins of attraction was utilised to characterise their fractal nature. Qualitative and quantitative results consistently confirm that the mechanism behind the sensitivity to initial conditions in this model originates from fractal basins of attraction, thereby distinguishing it from the commonly held belief that attributes it to chaos. This investigation provides valuable insights into the global dynamic behaviour and engineering applications of magnetic pendulums.