Automatic differentiation can replace hand-derived mathematics for discovering periodic orbits in chaotic systems, making it easier to find complex behaviors that were previously missed.
This paper develops an automated method to find and trace periodic orbits in dynamical systems like the double pendulum. Instead of hand-coding equations, it uses machine learning's automatic differentiation to optimize Fourier series representations of orbits, then follows flat directions in the loss landscape to efficiently discover new periodic motions—including previously unknown ones.