Circulant type formulas for the eigenvalues of linear network maps

Given an admissible map γ f for a homogeneous network N , it is known that the Jacobian D ( x ) around fully synchronous point = 0 … again . Motivated by this, we study spectra of linear maps networks. In particular, define so-called multipliers. These are (relatively small) matrices depend linearly on coefficients response function, and whose eigenvalues together make up spectrum corresponding map. More precisely, given finite set multipliers Λ l 1 k class networks C containing This furthermore closed under taking quotient networks, subnetworks disjoint unions. We then show any in those (a subset of) multipliers, with fixed multiplicities m independently (finite dimensional) phase space node. The all independent, which implies one may find simply expressing trace as combination traces will give examples where need not be constructed, but can determined looking at small also multiplicative respect to composition maps.