Variational Continuation for Double Pendulum Periodic Orbits

The researchers proposed a variational continuation method based on the Heisenberg matrix for numerically calculating periodic orbits in dynamical systems. This method parameterizes candidate periodic orbits as Fourier series, defines a loss function based on the deviation of the orbits from physical differential equations, and uses automatic differentiation to replace the previous manual derivation of the Jacobian matrix. The continuation direction is determined by the flat direction of the loss landscape, enabling efficient search without the need for an integrator. The study successfully demonstrated the continuation of complete double pendulum oscillations starting from a fixed point, revealed the bifurcation phenomenon of orbit families, and classified the branches. Additionally, the method identified two periodic orbits where the masses of the two pendulums were never simultaneously at rest, filling a gap in previous literature.