Similarity networks of ordinal-pattern transitions classify falling paper trajectories

Abstract: Paper fragments in free fall constitute a simple yet paradigmatic mechanical system exhibiting remarkably complex motions. Despite a long history of investigation, this system has defied comprehensive first-principles modeling, motivating the development of phenomenological and experimental approaches to classify the free-fall dynamics of small paper fragments. Here we apply the Bandt-Pompe symbolization method to extract high-dimensional features corresponding to ordinal-pattern transitions (so-called ordinal networks) from observed area time series of video-recorded falling papers shaped as circles, squares, hexagons, and crosses. We then represent each trajectory as a node in a weighted similarity network, with edges encoding pairwise dynamical similarity, and identify motion classes via community detection. Our method automatically clusters trajectories into tumbling and chaotic falls in excellent agreement with expert visual classification. Notably, it outperforms previous approaches based on classical physical features derived from complete three-dimensional trajectories -- especially for cross-shaped papers -- and requires no prior specification of the number of motion classes. We further find that trajectories diverging from expert classifications occupy more central positions in the similarity network, suggesting more complex and ambiguous dynamic behavior.