Hessian based method finds new periodic motions in double pendulums
Variational Continuation for Double Pendulum Periodic Orbits
Machine Learning
Summary
Finding repeating swinging patterns in systems like double pendulums can be tricky, especially when the motions are unstable or complex. The authors developed a computer approach that represents these patterns using waves and automatically checks how well they fit the physics, using techniques borrowed from machine learning. This method helps track how swinging motions change as conditions vary, detecting important changes and new types of motion. Using their approach, the authors found previously unknown periodic swinging patterns where both pendulum parts never stop moving at the same time. This offers fresh insights into the complicated behavior of double pendulums.
double pendulumperiodic orbitFourier seriesdynamical systemsautomatic differentiationHessian matrixloss functionbifurcationfixed pointssubharmonic bifurcation
Authors
Leo Yao, Ziming Liu, Max Tegmark
Abstract
We present a Hessian-based approach to numerically continue periodic orbits in dynamical systems. A loop (periodic orbit candidate) is parametrized as a Fourier series; a loss function is defined based on the deviation of the loop from the physical differential equations. Unlike previous work relying on hand-derived Jacobians, our method automates the process by leveraging automatic differentiation, a common machine learning technique. The continuation direction can be determined by the flat directions of the loss landscapes (directions with zero eigenvalues), making the search of periodic orbits efficient and guided. Our method is integrator-free, precisely initializes oscillations around unstable fixed points, and efficiently detects orbit family intersections and subharmonic bifurcations. As a demonstration, we present full continuations of periodic double pendulum oscillations from fixed points, showing bifurcations along orbit families and categorizing branches of periodic orbits. In particular, we find periodic orbits where both pendulum masses are never simultaneously at rest, which to our knowledge has been missing in the literature.