Feed-forward networks, center manifolds, and forcing

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 $\lambda^{\frac{1}{6}}$. We also use this center manifold structure to provide a method for classifying the bifurcation diagrams of the forced feed-forward chain where the amplitudes 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 bifurcations to three-cell feed-forward chains.