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
From the original article
1 min read at arxiv.orgWe 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.
By Leo Yao · arxiv.org
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