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For a regular coupled cell network, synchrony subspaces are the polydiagonalsubspaces that are invariant under the network adjacency matrix. The completelattice of synchrony subspaces of an $n$-cell regular network can be seen as anintersection of the partition lattice of $n$ elements and a lattice ofinvariant subspaces of the associated adjacency matrix. We assign integertuples with synchrony subspaces, and use them for identifying equivalentsynchrony subspaces to be merged. Based on this equivalence, the initiallattice of synchrony subspaces can be reduced to a lattice of synchronysubspaces which corresponds to a simple eigenvalue case discussed in ourprevious work. The result is a reduced lattice of synchrony subspaces, whichaffords a well-defined non-negative integer index that leads to bifurcationanalysis in regular coupled cell networks.

Reduced Lattices of Synchrony Subspaces and their Indices