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Understanding the behavior and evolution of a dynamical many-body system by analyzing patterns in their experimentally captured images is a promising method relevant for a variety of living and non-living self-assembled systems. The arrays of moving liquid crystal skyrmions studied here are a representative example of hierarchically organized materials that exhibit complex spatiotemporal dynamics driven by multiscale processes. Joint geometric and topological data analysis (TDA) offers a powerful framework for investigating such systems by capturing the underlying structure of the data at multiple scales. In the TDA approach, we introduce the $Ψ$-function, a robust numerical topological descriptor related to both the spatiotemporal changes in the size and shape of individual topological solitons and the emergence of regions with their different spatial organization. The geometric method based on the analysis of vector fields generated from images of skyrmion ensembles offers insights into the nonlinear physical mechanisms of the system's response to external stimuli and provides a basis for comparison with theoretical predictions. The methodology presented here is very general and can provide a characterization of system behavior both at the level of individual pattern-forming agents and as a whole, allowing one to relate the results of image data analysis to processes occurring in a physical, chemical, or biological system in the real world.

They're minute disruptions in the orientations of the molecules that make up solutions of liquid crystals, said Hayley Sohn, lead author of the new study. But under the microscope, these molecular deformations -- 10 of which could fill the width of a human hair -- certainly look alive. These pseudo-particles can twirl together as a group, shift their motion on a dime and even flow around obstacles when exposed to different electric currents. "By tuning that voltage, I can have them move in different directions and make them form a nice cluster where they're all stuck together. They can branch out into a chain and then come back together," said Sohn, a graduate student in the Materials Science and Engineering Program at CU Boulder.