Complexity Reduction Ansatz for Systems of Interacting Orientable Agents: Beyond The Kuramoto Model

TL;DR

This paper extends the reduction techniques used for phase oscillator systems like Kuramoto to more complex agents with higher-dimensional dynamics, enabling simplified analysis of their long-term behavior.

Contribution

It introduces a generalized model class of coupled agents with higher-dimensional states, demonstrating the existence of invariant manifolds for these systems.

Findings

Invariant manifolds exist for higher-dimensional agent systems

Reduced models facilitate analysis of complex interactions

Potential applications in diverse multi-agent scenarios

Abstract

Previous results have shown that a large class of complex systems consisting of many interacting heterogeneous phase oscillators exhibit an attracting invariant manifold. This result has enabled reduced analytic system descriptions from which all the long term dynamics of these systems can be calculated. Although very useful, these previous results are limited by the restriction that the individual interacting system components have one-dimensional dynamics, with states described by a single, scalar, angle-like variable (e.g., the Kuramoto model). In this paper we consider a generalization to an appropriate class of coupled agents with higher-dimensional dynamics. For this generalized class of model systems we demonstrate that the dynamics again contain an invariant manifold, hence enabling previously inaccessible analysis and improved numerical study, allowing a similar simplified description of these systems. We also discuss examples illustrating the potential utility of our results for a wide range of interesting situations.