A study of the double pendulum using polynomial optimization

TL;DR

This paper explores using polynomial optimization and sum-of-squares techniques to approximate safe initial conditions in a chaotic double pendulum system, providing computational methods to analyze complex dynamical behaviors.

Contribution

It demonstrates how polynomial barrier functions can be used to approximate sets of initial conditions that avoid certain behaviors in a chaotic system, specifically the double pendulum.

Findings

Semialgebraic sets closely approximate the safe initial conditions.

Computational methods effectively characterize chaotic system behaviors.

Polynomial optimization provides a viable approach for dynamical system analysis.

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.