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

Area of Science:

• Complex Systems
• Dynamical Systems Theory
• Mathematical Modeling

Background:

• Classical Kuramoto oscillators use a single phase variable.
• Swarmalators extend this to two variables: position and phase.
• Many natural systems exhibit interactions with three or more coupled variables.

Purpose of the Study:

• To develop a generalized oscillator model for systems with at least three mutually coupled state variables.
• To analyze the collective dynamics of such systems within a mathematically tractable framework.
• To explore the applicability of this model to diverse scientific domains.

Main Methods:

• Introduction of a novel oscillator model with three or more variables, each evolving on a circle.
• Mathematical analysis of system dynamics constrained to a 3-torus manifold (T³).
• Identification and characterization of distinct collective states and behaviors.

Main Results:

• The proposed model successfully accommodates systems with multiple coupled state variables.
• The 3-torus framework enables analytical treatment of complex collective behaviors.
• A variety of emergent collective dynamics were revealed.

Conclusions:

• The generalized oscillator model provides a powerful framework for understanding systems with intricate multi-variable interactions.
• This approach has broad applicability in fields such as sociology, collective animal motion, and neuroscience.
• The study opens new avenues for analyzing complex emergent phenomena in coupled systems.