Using curvature to understand the structure of dynamics

Abstract: In this talk, we will discuss the application of curvature to understanding the shape of dynamics of an attractor -- specifically features like unstable hyperbolic equilibrium points. Our primary focus, however, will be to demonstrate the use of curvature as an important tool in dynamical reconstruction of time-series data. We propose a curvature-based approach for choosing good values for the time-delay parameter tau in delay-coordinate reconstructions. The idea is based on exploiting the geometry of delay reconstructions. If the delay is chosen too small, the reconstructed dynamics are flattened along the main diagonal of the embedding space; too-large delays, on the other hand, can overfold those dynamics. Calculating the curvature of a two-dimensional delay reconstruction is an effective way to identify these extremes, and to find a middle ground between them, since both the sharp reversals at the ends of an insufficiently unfolded reconstruction and the folds in an overfolded one create spurious spikes in the curvature of a 2D projection of the reconstructed dynamics. We quantify this by computing various statistics over the Menger curvature of 2D reconstructions for different time delays. We discuss whether this result generalizes to higher-dimensional embeddings of the dynamics. Finally, we show that properties of the distribution of Menger curvature, such as its mean over the reconstructed trajectory as a function of tau, give an effective heuristic for choosing the time delay.