Nilpotent Hopf Bifurcations in Coupled Cell Systems

Abstract

Network architecture can lead to robust synchrony in coupled systems and, surprisingly, to codimension one bifurcations from synchronous equilibria at which the associated Jacobian is nilpotent. We prove three theorems concerning nilpotent Hopf bifurcations from synchronous equilibria to periodic solutions, where the critical eigenvalues have algebraic multiplicity two and geometric multiplicity one, and discuss these results in the context of three different networks in which the bifurcations occur generically. Phenomena stemming from these bifurcations include multiple periodic solutions, solutions that grow at a rate faster than the standard $\lambda^{\frac{1}{2}}$, and solutions that grow slower than the standard $\lambda^{\frac{1}{2}}$. These different bifurcations depend on the network architecture and, in particular, on the flow-invariant subspaces that are forced to exist by the architecture.