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Steady states are invaluable in the study of dynamical systems.High-dimensional dynamical systems, due to a separation of time-scales, oftenevolve towards a lower dimensional manifold $M$. We introduce an approach tolocate saddle points (and other fixed points) that utilizes gradient extremalson such a priori unknown (Riemannian) manifolds, defined by adaptively sampledpoint clouds, with local coordinates discovered on-the-fly through manifoldlearning. The technique, which efficiently biases the dynamical system along acurve (as opposed to exhaustively exploring the state space), requiresknowledge of a single minimum and the ability to sample around an arbitrarypoint. We demonstrate the effectiveness of the technique on the M\"uller-Brownpotential mapped onto an unknown surface (namely, a sphere). Previous workemployed a similar algorithmic framework to find saddle points using Newtontrajectories and gentlest ascent dynamics; we therefore also offer a briefcomparison with these methods.

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