Amplified Hopf bifurcations in feed-forward networks

TL;DR

This paper extends the understanding of Hopf bifurcations in feed-forward networks, showing that they generically produce branches of periodic solutions with amplitudes that grow at fractional powers of the bifurcation parameter.

Contribution

It applies a developed normal form method to a broader class of feed-forward chains, revealing generic amplified Hopf bifurcations with specific amplitude growth rates.

Findings

Hopf bifurcations generate periodic solutions with amplitudes like λ^{1/2}, λ^{1/6}, λ^{1/18}

Amplified bifurcations occur generically in feed-forward networks of arbitrary length

The results generalize previous findings limited to specific subclasses of networks

Abstract

In a previous paper, 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 $\lambda^{1/2}$, $\lambda^{1/6}$, $\lambda^{1/18}$, etc. Such amplified Hopf branches were previously found by others in a subclass of feed-forward networks with three cells, first under a normal form assumption and later by explicit computations. We explain here how these bifurcations arise generically in a broader class of feed-forward chains of arbitrary length.