Using Curvature to Understand the Structure of Dynamics

Explore the application of curvature in understanding dynamical systems and time-series data reconstruction in this 40-minute seminar from the Santa Fe Institute. Delve into the use of curvature for analyzing attractor dynamics, focusing on features like unstable hyperbolic equilibrium points. Learn about a novel curvature-based approach for selecting optimal time-delay parameters in delay-coordinate reconstructions. Discover how this method exploits the geometry of delay reconstructions to find a balance between insufficient unfolding and overfolding of the reconstructed dynamics. Examine the calculation of Menger curvature in 2D reconstructions and its effectiveness in identifying extremes in delay selection. Investigate the potential generalization of this approach to higher-dimensional embeddings and explore how properties of the Menger curvature distribution can serve as an effective heuristic for choosing time delays in dynamical reconstruction.