Multilators on a 3-Torus: A framework for high-dimensional coupled oscillators

The classical Kuramoto oscillator typically involves a single scalar state variable representing the phase of each oscillator. In contrast, swarmalators, an extension of the framework of Kuramoto oscillators, incorporate two coupled variables per unit: one describing spatial position and the other representing phase or orientation. Depending on the application, these state variables may differ in their nature and interpretation. However, many natural systems involve interactions among three or more coupled state variables, raising the question of how to effectively describe their collective dynamics. In this work, we introduce an oscillator model that accommodates at least three mutually coupled state variables, each of which evolves on a circle. Consequently, the dynamics of the system are constrained to a 3-torus manifold (T^{3}=S^{1}×S^{1}×S^{1}) that allows analytical treatment of the distinct collective states. This framework reveals a variety of collective behaviors and offers broad applicability to systems in sociology, collective animal motion, neuroscience, and beyond.