  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.
