Variational Continuation for Double Pendulum Periodic Orbits
Researchers introduce a Hessian-based, integrator-free method using automatic differentiation to continue periodic orbits in the double pendulum, uncovering previously unreported orbit families.
A new arXiv paper presents a variational, Hessian-based framework for numerically continuing periodic orbits in dynamical systems, parametrizing candidate loops as Fourier series and minimizing deviation from the governing differential equations. Automatic differentiation replaces hand-derived Jacobians, and flat directions of the loss landscape guide the continuation search. The method is demonstrated on the double pendulum, mapping bifurcations along orbit families, including periodic orbits where neither pendulum mass is ever simultaneously at rest.
- Fourier-parametrized loops plus automatic differentiation remove hand-derived Jacobians
- Flat loss-landscape directions guide continuation toward new periodic orbits
- Full double-pendulum continuations reveal bifurcations and subharmonic branches
- Method is integrator-free and initializes oscillations near unstable fixed points
Full article160 words · extracted from arxiv.org · click to collapse
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.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.05337