The realization of admissible graphs for coupled vector fields

TL;DR

This paper addresses the inverse problem of realizing any vector field as an admissible network dynamics by constructing graphs with specific properties, enabling analysis of synchronization and chimera states in coupled oscillators.

Contribution

It introduces a method to construct all admissible graphs for a given vector field and provides bounds on their number, advancing the understanding of network dynamics.

Findings

Any vector field can be realized as an admissible network dynamics.

A systematic procedure to construct all non-equivalent admissible graphs is provided.

Upper bounds on the number of admissible graphs are established.

Abstract

In a coupled network cells can interact in several ways. There is a vast literature from the last twenty years that investigates this interacting dynamics under a graph theory formalism, namely as a graph endowed with an input-equivalence relation on the set of vertices that enables a characterization of the admissible vector fields that rules the network dynamics. The present work goes in the direction of answering an inverse problem: for $n \geq 2$, any mapping on $\mathbb{R}^n$ can be realized as an admissible vector field for some graph with the number of vertices depending on (but not necessarily equal to) $n$. Given a mapping, we present a procedure to construct all non-equivalent admissible graphs, up to the appropriate equivalence relation. We also give an upper bound for the number of such graphs. As a consequence, invariant subspaces under the vector field can be investigated as the locus of synchrony states supported by an admissible graph, in the sense that a suitable graph can be chosen to realize couplings with more (or less) synchrony than another graph admissible to the same vector field. The approach provides in particular a systematic investigation of occurrence of chimera states in a network of van der Pol identical oscillators.