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.

Automated summary

In dynamical systems, constructing a barrier function satisfying certain inequalities on phase space ensures trajectories do not enter specified sets, with sum-of-squares conditions used for enforcing nonnegativity of polynomials computationally. Using the undamped double pendulum as an example, computations provide semialgebraic sets as close inner approximations to the fractal set of all such initial positions, studying their characteristics in chaotic systems.