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arXiv · 2106.13518

A study of the double pendulum using polynomial optimization

Jeremy P Parker·David Goluskin·Geoffrey M Vasil
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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.

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Jeremy P Parker, David Goluskin, Geoffrey M Vasil. 2021-06-25. A study of the double pendulum using polynomial optimization. https://doi.org/10.1063/5.0061316

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