Manuela Aguiar , University of Porto The Patterns of Synchrony of a Coupled Cell Network

This is joint work with Ana Dias (University of Porto). A coupled cell network consists of a finite set of nodes (the cells) and a finite number of arrows. The cells represent the individual dynamics and the arrows the interactions between the individuals. An equivalence relation on the set of nodes can be defined symbolizing the following; two nodes are in the same cell equivalence class if they represent individual dynamics with the same state space. Given an equivalence relation ⊲⊳ on the set of cells, refining the cell equivalence relation, choose a coloring of the cells by setting cells in the same equivalence class with the same color. The equivalence relation ⊲⊳ is balanced if given any two cells of the same color, there is a color-preserving bijection between the corresponding sets of cells that are coupled to them. A polydiagonal is a subspace of the phase space of a coupled cell system considering equalities of certain groups of cell coordinates. A polydiagonal is called a synchrony subspace if it is invariant by the flow of any coupled cell system admissible by the network structure. Stewart and Golubitsky presented a criterion, involving only the network structure, to caracterize the synchrony subspaces; by considering the balanced equivalence relations on the set of cells of the network. Stewart proved that the set of all balanced equivalence relations of a network forms a complete lattice. We give a characterization of this lattice.