Phase locking and multistability in the topological Kuramoto model on cell complexes

Area of Science:

• Complex systems
• Network science
• Mathematical physics

Background:

• Higher-order interactions significantly influence collective dynamics in oscillator networks.
• Existing synchronization models often simplify network topology, potentially missing crucial higher-order effects.

Purpose of the Study:

• To introduce and apply topological nonlinear Kirchhoff conditions to analyze phase-locked states in the topological Kuramoto model.
• To investigate the role of network topology and boundary structure in collective dynamics and multistability.

Main Methods:

• Developed the topological nonlinear Kirchhoff conditions for characterizing phase-locked states.
• Utilized generalized independent cycles and winding numbers to quantify phase dynamics.
• Applied the framework to various network structures including rings, Platonic solids, and regular simplices.

Main Results:

• Identified a universal rule: multistability arises only when boundaries possess a minimum of five elements.
• Demonstrated that independent winding numbers across different dimensions generate cascades of multistability.
• Established a direct link between cell complex topology, boundary features, and phase-locking behavior.

Conclusions:

• The topological Kuramoto model and nonlinear Kirchhoff conditions provide a robust framework for understanding collective dynamics on cell complexes.
• Network topology and boundary structure are critical determinants of phase locking and multistability in complex systems.
• The findings offer a generalized approach applicable to diverse systems exhibiting higher-order interactions.