Abstract
Modeling the complex collective behavior is a challenging issue in several material and life sciences. The collective motion has been usually modeled by simple interaction rules and explained by global statistics. However, it remains difficult to bridge the gap between the dynamic properties of the complex interaction and the emerging group-level functions. Here we introduce decomposition methods to directly extract and classify the latent global dynamics of nonlinear dynamical systems in an equation-free manner, even including complex interaction in few data dimensions. We first verified that the basic decomposition method can extract and discriminate the dynamics of a well-known rule-based fish-schooling (or bird-flocking) model. The method extracted different temporal frequency modes with spatial interaction coherence among three distinct emergent motions, whereas these wave properties in multiple spatiotemporal scales showed similar dispersion relations. Second, we extended the basic method to map high-dimensional feature space for application to actual small-dimensional systems complexly changing the interaction rules. Using group sports human data, we classified the dynamics and predicted the group objective achievement. Our methods have a potential for classifying collective motions in various domains which obey in non-trivial dominance law known as active matters.