A study of the double pendulum using polynomial optimization

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 the 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.