On experimentally locating saddle-points on a potential energy surface from observed dynamics

Abstract

Abstract This paper details a new method to estimate the location of unstable equilibria, specifically saddle-points, based on transient trajectories from experiments. We describe a system in which saddle-points (not easily observed in a direct sense) influence the behavior of trajectories that pass ’close-by’ them. This influence is used to construct a model and thus identify a more accurate estimate of the location using a number of refinements associated with linearization and regression. Both simulations and experiments were conducted to verify the method. The experiment consists of a small ball rolling on a relatively shallow curved surface under the influence of gravity: a potential energy surface in two dimensions. Tracking the motion of the ball with a digital camera provides data that compares closely with the output of numerical simulation. The experimental results suggest that this method can effectively locate the saddle equilibria in a system, and the robustness of the approach is assessed relative to the effect of noise, size of the local neighborhood, etc., in addition to providing information on the local dynamics. Given the relative simplicity of the experiment system used and a priori knowledge of the saddle-points, it is a useful testing environment for system identification in a nonlinear context.