Path-dependency and emergent computing under vectorial driving

Abstract

The sequential response of frustrated materials-ranging from crumpled sheets and amorphous media to metamaterials-reveals their memory effects and emergent computational potential. Despite their spatial extension, most studies rely on a single global stimulus, such as compression, effectively reducing the problem to scalar driving. Here, we introduce vectorial driving of frustrated materials by applying multiple spatially localized stimuli to explore path-dependent, sequential responses. We uncover a wealth of phenomena absent in scalar driving, including non-Abelian responses, mixed-mode behavior, and chiral loop transients. We show that fold singularities connect three states -- ancestor, descendant, and sibling. This recurring pattern serves as the elementary building block of all sequential paths. We then introduce three levels of description of sequential, path-dependent responses. At the most fundamental level, path-dependent transition graphs (pt-graphs) and strain maps capture the response under arbitrary vectorial driving and connect pathways to the underlying singularities. They provide a complete description analogous to transition graphs (t-graphs) for scalar driving. However, as pt-graphs and strain maps become unwieldy for high-dimensional driving, we introduce b-graphs -- graphs whose nodes and transitions encode the systems response to binarized vectorial driving. These present a less complete but much simpler second-level description by restricting attention to binary input and their induced transitions. Finally, we introduce graph-based motifs that enable a systematic analysis of b-graphs. As statistical measures of pathway complexity, these motifs can be obtained in systems of any size or complexity. Our work paves the way for strategies to explore, harness, and understand complex materials and memory, while advancing embodied intelligence and in-materia computing.