In this talk, we consider rigid properties of hyperbolic periodic solutions of dynamical systems of networks. Let X_0=(x^0_1, ... , x^0_n) be a hyperbolic periodic solution to an admissible dynamical system of a network G. Let T be the minimal period of X_0. Cells (nodes) i, j are phase-related on X_0 if there exists a real number theta ( 0 <= theta < 1), such that x^0_i(t)= x^0_j(t+ theta T). The phase relation is rigid if the same phase relation remains in the perturbed periodic solution of any sufficiently small admissible perturbation. When theta =0, the two cells are called rigid synchrony. Rigid phase relation is known to be a product of symmetry. However, Stewart and Parker found that rigid phase relation might occur on a network without symmetry, but one of its quotient network defined by collapsing all rigid synchrony cells to one cell has symmetry. Stewart and Parker conjectured that this is the only way to get rigid phase relations on a transitive network without symmetry.
Stewart et. al. reduced the proving of this conjecture to the proving of the following two other conjectures:
1) phase rigid property: Suppose phase relations on X_0 are rigid, then for each pair of phase-related cells, the signals they receive are also phase-related with the same phase-relation.
2) fully oscillatory property: In a transitive network, a hyperbolic periodic solution of an admissible vector field of the network is generically fully oscillatory (all cells are oscillatory on the periodic solution).
We show that these two conjectures are actually correct. These results are joint work with Marty Golubitsky and David Romano.