M. Golubitsky and M. Krupa
Stability computations for nilpotent Hopf bifurcations in coupled cell systems
International Journal of Bifurcation and Chaos.
17 (2007) 2595-2603.
Vanderbauwhede and van Gils, Krupa, and Langford studied unfoldings of
bifurcations with purely imaginary eigenvalues and a nonsemisimple
linearization, which generically occurs in codimension three. In
networks of identical coupled ODE these nilpotent Hopf bifurcations
can occur in codimension one. Elmhirst and Golubitsky show that these
bifurcations can lead to surprising branching patterns of periodic
solutions, where the type of bifurcation depends in part on the
existence of an invariant subspace corresponding to partial synchrony.
We study the stability of some of these bifurcating solutions. In the
absence of partial synchrony the problem is similar to the generic
codimension three problem. In this case we show that the bifurcating
branches are generically unstable. When a synchrony subspace is present
we obtain partial stability results by using only those near identity
transformations that leave this subspace invariant.