Classification of Networks with Asymmetric Inputs

Abstract

Coupled cell systems associated with a coupled cell network are determined by (smooth) vector fields that are consistent with the network structure. Here, we follow the formalisms of Stewart, Golubitsky and Pivato (Symmetry groupoids and patterns of synchrony in coupled cell networks, SIAM J. Appl. Dyn. Syst. 2 (4) (2003) 609-646), Golubistky, Stewart and Török (Patterns of synchrony in coupled cell networks with multiple arrows, SIAM J. Appl. Dynam. Sys. 4 (1) (2005) 78-100) and Field (Combinatorial dynamics, Dynamical Systems 19 (2004) (3) 217-243). It is known that two non-isomorphic n-cell coupled networks can determine the same sets of vector fields -these networks are said to be ODE-equivalent. The set of all n-cell coupled networks is so partitioned into classes of ODE-equivalent networks. With no further restrictions, the number of ODE-classes is not finite and each class has an infinite number of networks. Inside each ODE-class we can find a finite subclass of networks that minimize the number of edges in the class, called minimal networks.In this paper, we consider coupled cell networks with asymmetric inputs. That is, if k is the number of distinct edges types, these networks have the property that every cell receives k inputs, one of each type. Fixing the number n of cells, we prove that: the number of ODE-classes is finite; restricting to a maximum of n(n − 1) inputs, we can cover all the ODE-classes; all minimal n-cell networks with n(n − 1) asymmetric inputs are ODE-equivalent. We also give a simple criterion to test if a network is minimal and we conjecture lower estimates for the number of distinct ODE-classes of n-cell networks with any number k of asymmetric inputs. Moreover, we present a full list of representatives of the ODE-classes of networks with three cells and two asymmetric inputs.

Steady-state bifurcations for three-cell networks with asymmetric inputs

“…✸ Remark 4.2. The minimal network in Figure 2 represents the unique ODE-class of minimal three-cell networks with six asymmetric inputs [7]. The Jacobian matrix of f N at the origin for such network has the form J N f = f 0 Id 3 + 6 i=1 f i A i , where A i , for i = 1, .…”

“…, 6 are the network adjacency matrices. From the results of Aguiar, Dias and Soares [7], we know that the six adjacency matrices A i plus the 3 × 3 identity matrix generate the linear space of the 3 × 3 matrices with constant row sum. It follows then that, besides the valency eigenfunction, the other eigenvalues of J N f are generically arbitrary and simple.…”

“…We then restrict to the three-cell networks with any number of asymmetric inputs, and finally to three-cell networks with one, two or six asymmetric inputs. This work is a follow-up of the work in Aguiar, Dias and Soares [7] where, using the results on the characterization of minimal networks by Aguiar and Dias [5], it is given a methodology for classifying identical cell networks with asymmetric inputs. In particular, it is shown that an ncell network of identical cells with asymmetric inputs is ODE-equivalent to an n-cell network of identical cells with at most n(n − 1) asymmetric inputs and that there is an unique ODE-class of identical cell networks with n(n − 1) asymmetric inputs.…”

“…Here, we study the codimension-one steady-state bifurcation problems from a full synchrony equilibrium for the networks enumerated in Aguiar, Dias and Soares [7]. The Jacobian of a coupled cell system associated with a network of identical cells with k asymmetric inputs at a full synchrony equilibrium is determined by the associated adjacency matrices.…”

“…Therefore those results show which synchrony spaces support (or not) bifurcating branches from a codimension-one steady-state bifurcations for generic coupled cell systems associated with networks having such synchrony lattice structures. In particular, we classify the synchrony spaces that support (or not) bifurcating branches for the networks enumerated in Aguiar, Dias and Soares [7].…”