A study of the double pendulum using polynomial optimization
Abstract
In dynamical systems governed by differential equations, a guarantee that trajectories emanating from a given set of initial conditions do not enter another given set can be obtained by constructing a barrier function that satisfies certain inequalities on phase space. Often these inequalities amount to nonnegativity of polynomials and can be enforced using sum-of-squares conditions, in which case barrier functions can be constructed computationally using convex optimization over polynomials. To study how well such computations can characterize sets of initial conditions in a chaotic system, we use the undamped double pendulum as an example and ask which stationary initial positions do not lead to flipping of the pendulum within a chosen time window. Computations give semialgebraic sets that are close inner approximations to the fractal set of all such initial positions.
Cite
@article{arxiv.2106.13518,
title = {A study of the double pendulum using polynomial optimization},
author = {Jeremy P Parker and David Goluskin and Geoffrey M Vasil},
journal= {arXiv preprint arXiv:2106.13518},
year = {2022}
}