A novel global perspective: Characterizing the fractal basins of attraction and the level of chaos in a double pendulum

The objective of this work is to deeply investigate the sensitivity to initial conditions and the factors influencing the level of chaos in a double pendulum system from a novel global perspective. Firstly, the pendulum's motion trajectories and mechanical energy are compared to determine the appropriate numerical algorithms for solving this model, including the fourth-order Runge-Kutta method (RK4 method) and the Euler method. Secondly, the captured experimental motion trajectories, along with numerical results, vividly demonstrate the system's sensitivity to initial conditions. On this basis, we develop an algorithm that successfully delineates the basins of attraction associated with the number of flips and the final angular positions of the pendulum, uncovering a petal-like structure characterized by significant rotational symmetry and fractal features. Finally, we employ a heat map of the average maximum Lyapunov exponent to reveal the correlation between mass ratio and the level of chaos. Both qualitative and quantitative results consistently confirm the mechanisms underlying the system's sensitivity to initial conditions and the reliability of the developed algorithm. This research provides valuable insights into the global dynamics and engineering applications of the double pendulum system.