Exploring the potential energy landscape of the Thomson problem via Newton homotopies

Locating the stationary points of a real-valued multivariate potential energy function is an important problem in many areas of science. This task generally amounts to solving simultaneous nonlinear systems of equations. While there are several numerical methods that can find many or all stationary points, they each exhibit characteristic problems. Moreover, traditional methods tend to perform poorly near degenerate stationary points with additional zero Hessian eigenvalues. We propose an efficient and robust implementation of the Newton homotopy method, which is capable of quickly sampling a large number of stationary points of a wide range of indices, as well as degenerate stationary points. We demonstrate our approach by applying it to the Thomson problem. We also briefly discuss a possible connection between the present work and Smale's 7th problem.