Shooting the Death Star: An Exploration of the Double Pendulum

TL;DR: Check out the file heatmap_tau100.png below for a map of the double pendulum's divergence depending on initial conditions. Read Shooting_the_Death_Star.pdf for the paper. This paper tackles three linked problems around the canonical chaotic system of the double pendulum. First, it benchmarks ODE solvers and JIT-accelerated variants, finding DOP853 to be the most reliable and efficient choice within SciPy for non-stiff, high-accuracy integrations; LSODA can be competitive but erratic, and Euler is computationally ineffective; Numba/NumbalSODA provide modest to ~5× speedups with trade-offs, so the study proceeds with DOP853 for simplicity. Second, it proposes a practical finite-time divergence metric (first-passage of Euclidean separation) and generates high-resolution (1800×1800) maps over initial angles showing where perturbations grow by 10×/100×//1,000×/10,000×. These maps reveal strands of chaos penetrating the “no-flip” region, stable islands with likely fractal boundaries, and a clear distinction between instability of initial equilibria and sustained chaotic dynamics. Third, it generalizes the model to distributed masses with damping and applies it to a rotating “Death Star” shooting target. An adaptive Monte-Carlo across shooter speed/accuracy and decision rules shows a steep learning cliff: performance collapses below roughly 0.8 s shot-cycle or ~6 cm (1σ) accuracy. Optimal tactics depend on skill: fast shooters should wait for lower plate speeds, while slower shooters do better engaging liberally. Selecting the fastest plate (and, second-best, rebalancing) outperforms common heuristics like “topmost” or “slowest.” This paper was written for the CAS-522: Methods for Complex Systems Science—Dynamical Systems course at Arizona State University in the fall semester of 2025.