Autodiff Replaces Hand-Derived Jacobians for Continuing Periodic Orbits, Demonstrated on the Double Pendulum

arXiv 2609.05337·low signalXBlueskyLinkedInCopy link

arXiv 2609.05337, from Leo Yao, Ziming Liu and Max Tegmark, parametrizes a periodic orbit candidate as a Fourier series and defines a loss from the loop's deviation from the governing differential equations, then continues orbits using a Hessian-based method. The contribution over prior work is dropping hand-derived Jacobians in favor of automatic differentiation, with the continuation direction picked from the flat, zero-eigenvalue directions of the loss landscape. The method is integrator-free, initializes oscillations around unstable fixed points precisely, and detects orbit family intersections and subharmonic bifurcations, demonstrated on full double pendulum continuations.