Towards a classification of networks with asymmetric inputs

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