Linear Equivalence and ODE-equivalence for Coupled Cell Networks

Coupled cell systems are systems of ODEs, defined by ‘admissible’ vector fields, associated with a network whose nodes represent variables and whose edges specify couplings between nodes. It is known that non-isomorphic networks can correspond to the same space of admissible vector fields. Such networks are said to be ‘ODE-equivalent’. We prove that two networks are ODE-equivalent if and only if they determine the same space of linear vector fields; moreover, the variable associated with each node may be assumed 1-dimensional for that purpose. We briefly discuss the combinatorics of the resulting linear algebra problem. AMS classification scheme numbers: 37C10 20L05