Amplified Hopf Bifurcations in Feed-Forward Networks

In [18] the authors developed a method for computing normal forms of dynamical systems with a coupled cell network structure. We now apply this theory to one-parameter families of homogeneous feed-forward chains with 2-dimensional cells. Our main result is that Hopf bifurcations in such families generically generate branches of periodic solutions with amplitudes growing like ∼ |λ| 1 2 ,∼ |λ| 1 6 ,∼ |λ| 1 18 , etc. Such amplified Hopf branches were previously found in a subclass of feed-forward networks with three cells, first under a normal form assumption [15] and later by explicit computations [8], [13]. We explain here how these bifurcations arise generically in a broader class of feed-forward chains of arbitrary length.