Complexity reduction ansatz for systems of interacting orientable agents: Beyond the Kuramoto model

Area of Science:

• Complex Systems
• Nonlinear Dynamics
• Statistical Physics

Background:

• Previous studies identified attracting invariant manifolds in complex systems of interacting phase oscillators.
• These manifolds allowed reduced analytic descriptions for systems with one-dimensional dynamics, exemplified by the Kuramoto model.
• Existing methods were limited to systems with scalar state variables.

Purpose of the Study:

• To generalize the concept of invariant manifolds to coupled agents with higher-dimensional dynamics.
• To enable reduced analytic system descriptions for a broader class of complex systems.
• To facilitate improved numerical study and analysis of these generalized systems.

Main Methods:

• Mathematical analysis of coupled agent systems with higher-dimensional dynamics.
• Demonstration of the existence of an invariant manifold in these generalized systems.
• Development of reduced analytic descriptions based on the identified invariant manifold.

Main Results:

• The existence of an attracting invariant manifold was proven for a generalized class of coupled agents with higher-dimensional dynamics.
• A simplified analytic description analogous to one-dimensional systems was achieved.
• The findings enable previously inaccessible analysis and improved numerical studies.

Conclusions:

• The generalization of invariant manifolds to higher-dimensional dynamics significantly expands the applicability of reduced-order modeling in complex systems.
• These results offer a powerful tool for analyzing and understanding the long-term behavior of a wider range of heterogeneous interacting systems.
• The study highlights the potential utility for diverse applications in physics, engineering, and beyond.