M. Golubitsky, D. Romano and Y. Wang
Network periodic solutions: full oscillation and rigid synchrony
Nonlinearity.
23 (2010) 3227-3243.
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 differentialequations
oscillate in each node if and only if the network is transitive.
We can associate a polydiagonal Delta(Z(t)) to each hyperbolic periodic
solution Z(t) as follows. The cell coordinates of a pointin Delta(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 Delta(Z(t)) is rigid (robust to small admissible perturbations)
if only if it is flow-invariant for all admissible vector fields.