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

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.

arXiv comment8 pages (incl. 1 appendix), 2 figures

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• The Kuramoto model on a sphere: Explaining its low-dimensional dynamics with group theory and hyperbolic geometry