Network Periodic Solutions: Full Oscillation and Rigid Synchrony

We prove two results about hyperbolic periodic solutions in networks of systems of ODEs. First, we show that generically hyperbolic periodic solutions of network admissible systems of differential equations oscillate in each node if and only if the network is transitive. We can associate a polydiagonal ∆(Z(t)) to each hyperbolic periodic solution Z(t) as follows. The cell coordinates of a point in ∆(Z(t)) are equal if the corresponding cell coordinates of Z(t) are equal for all t; that is, the output from the two cells are synchronous. Second, we prove that ∆(Z(t)) is rigid (robust to small admissible perturbations) if only if it is flow-invariant for all admissible vector fields.