M. Golubitsky and C. Postlethwaite
Feed-forward networks, center manifolds, and forcing
Discrete and Continuous Dynamical Systems - Series A.
32 (2012) 2913-2935.
This paper discusses feed-forward chains near points of synchrony-breaking Hopf bifurcation.
We show that at synchrony-breaking bifurcations the center manifold inherits a feed-forward
structure and use this structure to provide a simplified proof of the theorem of Elmhirst
and Golubitsky that there is a branch of periodic solutions in such bifurcations whose
amplitudes grow at the rate of lambda^{1/6}. We also use this center manifold structure to
provide a method for classifying the bifurcation diagrams of the forced feed-forward chain
where the amplitude of the periodic responses are plotted as a function of the forcing
frequency. The bifurcation diagrams depend on the amplitude of the forcing, the deviation of
the system from Hopf bifurcation, and the ratio gamma of the imaginary part of the cubic term
in the normal form of Hopf bifurcation to the real part. These calculations generalize the
results of Zhang on the forcing of systems near Hopf bifurcation to three-cell feed-forward chains.