Michael Field and James Swift
Hopf bifurcation and the Hopf fibration
We present techniques for studying the local dynamics generated by
an equivariant Hopf bifurcation. We show that under general hypotheses
we can expect the formation of a branch of attracting invariant spheres
which capture all the local dynamics. In addition, using the Hopf fibration,
we show that the limit cycles generated by the Hopf bifurcation are
determined by zeros of a vector field
defined on complex projective space. We show how to compute these
zeros and illustrate our methods with examples of Hopf bifurcations for the
dihedral groups of order six and eight and the orthogonal groups.