Amplified Hopf Bifurcations in Feed-Forward Networks

Abstract

In [B. Rink and J. Sanders, Trans. Amer. Math. Soc., to appear] 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 $\sim|\lambda|^{\frac{1}{2}},\sim|\lambda|^{\frac{1}{6}},\sim|\lambda|^{\frac{1}{18}}$, etc. Such amplified Hopf branches were previously found in a subclass of feed-forward networks with three cells, first under a normal form assumption [M. Golubitsky and I. Stewart, Bull. Amer. Math. Soc. (N.S.), 43 (2006), pp. 305--364] and later by explicit computations [T. Elmhirst and M. Golubitsky, SIAM J. Appl. Dyn. Syst., 5 (2006), pp. 205--251], [M. Golubitsky and C. Postlethwaite, Discrete Contin. Dyn. Syst., 32 (2012), pp. 2913--2935]. We explain here how these bifurcations arise generically in a broader class of feed-forward chains of arbitrary length.