Gradient systems on coupled cell networks

TL;DR

This paper characterizes admissible gradient functions on coupled cell networks using graph theory, analyzing equilibrium patterns and their relation to the network's topological structure.

Contribution

It provides a general form of admissible functions for gradient networks and links their critical points to the graph's spectral properties, including specific equilibrium configurations.

Findings

Characterized admissible functions via graph topology.

Linked equilibria to spectral properties of the graph.

Analyzed synchronous and 2-state equilibrium patterns.

Abstract

For networks of coupled dynamical systems we characterize admissible functions, that is, functions whose gradient is an admissible vector field. The schematic representation of a gradient network dynamical system is of an undirected cell graph, and we use tools from graph theory to deduce the general form of such functions, relating it to the topological structure of the graph defining the network. The coupling of pairs of dynamical systems cells is represented by edges of the graph, and from spectral graph theory we detect the existence and nature of equilibria of the gradient system from the critical points of the coupling function. In particular, we study fully synchronous and 2-state patterns of equilibria on regular graphs.These are two special types of equilibrium configurations for gradient networks. We also investigate equilibrium configurations of S1-invariant admissible functions on a ring of cells.