Gradient descent algorithm to search for periodic [formula omitted]-body orbits

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We describe a gradient-descent based algorithm to compute approximate periodic orbits of an N-body system. Given initial position and velocity vectors, the N-body system is numerically evolved using a Runge–Kutta solver which is end-to-end differentiable, such that loss gradients can be computed. The frame of the trajectories which is most similar to the starting conditions, approximately one orbital period after t=0, is computed, and the L1 norm between the initial conditions and the conditions of this frame is minimized. In this manner, the trajectories are optimized to exhibit periodic behavior. We apply this algorithm to the three-body problem to find quasi-stable orbits, but this algorithm can be applied to other chaotic systems.