  We continue our investigation of versality for parametrized families of
linear retarded functional differential equations (RFDEs) projected onto
finite-dimensional invariant manifolds. In this paper, we consider RFDEs
equivariant with respect to the action of a compact Lie group. In a previous
paper (Buono and LeBlanc, to appear in J. Diff. Eqs.), we have studied this
question in the general case (i.e. no a priori restrictions on the RFDE). When
studying the question of versality in the equivariant context, it is natural to
want to restrict the range of possible unfoldings to include only those which
share the same symmetries as the original RFDE, and so our previous results do
not immediately apply. In this paper, we show that with appropriate
projections, our previous results on versal unfoldings of linear RFDEs can be
adapted to the case of linear equivariant RFDEs. We illustrate our theory by
studying the linear equivariant unfoldings at double Hopf bifurcation points in
a D_3-equivariant network of coupled identical neurons modeled by
delay-differential equations due to delays in the internal dynamics and
coupling.
