7 research outputs found
Amplified Hopf bifurcations in feed-forward networks
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 , , , 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
Normal-normal resonances in a double Hopf bifurcation
We investigate the stability loss of invariant n-dimensional quasi-periodic
tori during a double Hopf bifurcation, where at bifurcation the two normal
frequencies are in normal-normal resonance. Invariants are used to analyse the
normal form approximations in a unified manner. The corresponding dynamics form
a skeleton for the dynamics of the original system. Here both normal
hyperbolicity and KAM theory are being used.Comment: 22 pages, 6 figure
Normal resonances in a double Hopf bifurcation
We introduce a framework to systematically investigate the resonant double Hopf bifurcation. We use the basic invariants of the ensuing T1-action to analyse the approximating normal form truncations in a unified manner. In this way we obtain a global description of the parameter space and thus find the organising resonance droplet, which is the present analogue of the resonant gap. The dynamics of the normal form yields a skeleton for the dynamics of the original system. In the ensuing perturbation theory both normal hyperbolicity (centre manifold theory) and KAM theory are being used
Center manifolds of coupled cell networks
Dynamical systems with a network structure can display anomalous bifurcations
as a generic phenomenon. As an explanation for this it has been noted that
homogeneous networks can be realized as quotient networks of so-called
fundamental networks. The class of admissible vector fields for these
fundamental networks is equal to the class of equivariant vector fields of the
regular representation of a monoid. Using this insight, we set up a framework
for center manifold reduction in fundamental networks and their quotients. We
then use this machinery to explain the difference in generic bifurcations
between three example networks with identical spectral properties and identical
robust synchrony spaces