  We study stability of so-called synchronous slowly oscillating periodic
solutions (SOPSs) for a system of identical delay differential equations (DDEs)
with linear decay and nonlinear delayed negative feedback that are coupled
through their nonlinear term. Under a row sum condition on the coupling matrix,
existence of a unique SOPS for the corresponding scalar DDE implies existence
of a unique synchronous SOPS for the coupled DDEs. However, stability of the
SOPS for the scalar DDE does not generally imply stability of the synchronous
SOPS for the coupled DDEs. We obtain an explicit formula, depending only on the
spectrum of the coupling matrix, the strength of the linear decay and the
values of the nonlinear negative feedback function near plus/minus infinity,
that determines the stability of the synchronous SOPS in the asymptotic regime
where the nonlinear term is heavily weighted. We also treat the special cases
of so-called weakly coupled systems, near uniformly coupled systems, and doubly
nonnegative coupled systems, in the aforementioned asymptotic regime. Our
approach is to estimate the characteristic (Floquet) multipliers for the
synchronous SOPS. We first reduce the analysis of the multidimensional
variational equation to the analysis of a family of scalar variational-type
equations, and then establish limits for an associated family of monodromy-type
operators. We illustrate our results with examples of systems of DDEs with
mean-field coupling and systems of DDEs arranged in a ring.
