Variational Continuation for Double Pendulum Periodic Orbits

Finding periodic orbits in chaotic dynamical systems traditionally demands tedious manual derivation of Jacobians or sensitive numerical integration. By parameterizing candidate orbits with Fourier series and applying automatic differentiation to construct an integrator-free loss landscape, this work identifies continuation trajectories through Hessian flat directions. Demonstrating the approach on the classic double pendulum, the authors mapped out bifurcations and family branches from fixed points, uncovering periodic orbits where neither pendulum mass comes to rest at the same instant.