Low-dimensional dynamics in non-Abelian Kuramoto model on the 3-sphere

Area of Science:

• Complex systems
• Non-Abelian dynamics
• Mathematical physics

Background:

• The Kuramoto model describes coupled oscillators, widely used in physics and neuroscience.
• Generalizing the Kuramoto model to non-Abelian groups and spheres introduces complex dynamics.
• Understanding low-dimensional dynamics is crucial for simplifying complex systems.

Purpose of the Study:

• To investigate the low-dimensional dynamics of generalized oscillators in the non-Abelian Kuramoto model on the 3-sphere.
• To identify the governing equations and transformations for these oscillators.
• To explore the impact of uniform initial distributions on system symmetries.

Main Methods:

• Analysis of the general non-Abelian Kuramoto model with generalized oscillators.
• Identification of global variables determining system dynamics under specific conditions.
• Derivation of quaternion-valued ordinary differential equations governing global variables.
• Investigation of symmetries arising from uniform initial distributions.

Main Results:

• The dynamics of identical generalized oscillators with global coupling are determined by a few global variables.
• These oscillators evolve under the action of a specific group of (quaternionic) Möbius transformations.
• A system of quaternion-valued ODEs, extending the Watanabe-Strogatz system, governs the global variables.
• Uniform initial distributions lead to additional symmetries, restricting dynamics to invariant submanifolds.

Conclusions:

• The non-Abelian Kuramoto model on the 3-sphere exhibits low-dimensional dynamics under specific conditions.
• The identified transformations and ODEs provide a simplified framework for analyzing these complex systems.
• The emergence of symmetries under uniform distributions offers further avenues for dynamical reduction.