Stability Computations for Nilpotent Hopf bifurcations in Coupled Cell Systems

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 showed 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.