Locating saddle points using gradient extremals on manifolds adaptively revealed as point clouds

Steady states are invaluable in the study of dynamical systems. High-dimensional dynamical systems, due to a separation of time-scales, often evolve towards a lower dimensional manifold $M$. We introduce an approach to locate saddle points (and other fixed points) that utilizes gradient extremals on such a priori unknown (Riemannian) manifolds, defined by adaptively sampled point clouds, with local coordinates discovered on-the-fly through manifold learning. The technique, which efficiently biases the dynamical system along a curve (as opposed to exhaustively exploring the state space), requires knowledge of a single minimum and the ability to sample around an arbitrary point. We demonstrate the effectiveness of the technique on the M\"uller-Brown potential mapped onto an unknown surface (namely, a sphere). Previous work employed a similar algorithmic framework to find saddle points using Newton trajectories and gentlest ascent dynamics; we therefore also offer a brief comparison with these methods.